Elementary Functions Part 1, Functions Lecture 1.4a, Symmetries of - - PowerPoint PPT Presentation

Elementary Functions

Part 1, Functions Lecture 1.4a, Symmetries of Functions: Even and Odd Functions

• Dr. Ken W. Smith

Sam Houston State University

2013

Smith (SHSU) Elementary Functions 2013 1 / 25

Even and odd functions

In this lesson we look at even and odd functions. A symmetry of a function is a transformation that leaves the graph unchanged. Consider the functions f(x) = x2 and g(x) = |x| whose graphs are drawn below. Both graphs allow us to view the y-axis as a mirror. A reflection across the y-axis leaves the function unchanged. This reflection is an example of a symmetry.

Smith (SHSU) Elementary Functions 2013 2 / 25

Even and odd functions

In this lesson we look at even and odd functions. A symmetry of a function is a transformation that leaves the graph unchanged. Consider the functions f(x) = x2 and g(x) = |x| whose graphs are drawn below. Both graphs allow us to view the y-axis as a mirror. A reflection across the y-axis leaves the function unchanged. This reflection is an example of a symmetry.

Smith (SHSU) Elementary Functions 2013 2 / 25

Even and odd functions

In this lesson we look at even and odd functions. A symmetry of a function is a transformation that leaves the graph unchanged. Consider the functions f(x) = x2 and g(x) = |x| whose graphs are drawn below. Both graphs allow us to view the y-axis as a mirror. A reflection across the y-axis leaves the function unchanged. This reflection is an example of a symmetry.

Smith (SHSU) Elementary Functions 2013 2 / 25

Even and odd functions

In this lesson we look at even and odd functions. A symmetry of a function is a transformation that leaves the graph unchanged. Consider the functions f(x) = x2 and g(x) = |x| whose graphs are drawn below. Both graphs allow us to view the y-axis as a mirror. A reflection across the y-axis leaves the function unchanged. This reflection is an example of a symmetry.

Smith (SHSU) Elementary Functions 2013 2 / 25

Even and odd functions

In this lesson we look at even and odd functions. A symmetry of a function is a transformation that leaves the graph unchanged. Consider the functions f(x) = x2 and g(x) = |x| whose graphs are drawn below. Both graphs allow us to view the y-axis as a mirror. A reflection across the y-axis leaves the function unchanged. This reflection is an example of a symmetry.

Smith (SHSU) Elementary Functions 2013 2 / 25

Even and odd functions

In this lesson we look at even and odd functions. A symmetry of a function is a transformation that leaves the graph unchanged. Consider the functions f(x) = x2 and g(x) = |x| whose graphs are drawn below. Both graphs allow us to view the y-axis as a mirror. A reflection across the y-axis leaves the function unchanged. This reflection is an example of a symmetry.

Smith (SHSU) Elementary Functions 2013 2 / 25

Reflection across the y-axis

A symmetry of a function can be represented by an algebra statement. Reflection across the y-axis interchanges positive x-values with negative x-values, swapping x and −x. Therefore f(−x) = f(x). The statement, “For all x ∈ R, f(−x) = f(x)” is equivalent to the statement “The graph of the function is unchanged by reflection across the y-axis.”

Smith (SHSU) Elementary Functions 2013 3 / 25

Reflection across the y-axis

A symmetry of a function can be represented by an algebra statement. Reflection across the y-axis interchanges positive x-values with negative x-values, swapping x and −x. Therefore f(−x) = f(x). The statement, “For all x ∈ R, f(−x) = f(x)” is equivalent to the statement “The graph of the function is unchanged by reflection across the y-axis.”

Smith (SHSU) Elementary Functions 2013 3 / 25

Reflection across the y-axis

A symmetry of a function can be represented by an algebra statement. Reflection across the y-axis interchanges positive x-values with negative x-values, swapping x and −x. Therefore f(−x) = f(x). The statement, “For all x ∈ R, f(−x) = f(x)” is equivalent to the statement “The graph of the function is unchanged by reflection across the y-axis.”

Smith (SHSU) Elementary Functions 2013 3 / 25

Reflection across the y-axis

A symmetry of a function can be represented by an algebra statement. Reflection across the y-axis interchanges positive x-values with negative x-values, swapping x and −x. Therefore f(−x) = f(x). The statement, “For all x ∈ R, f(−x) = f(x)” is equivalent to the statement “The graph of the function is unchanged by reflection across the y-axis.”

Smith (SHSU) Elementary Functions 2013 3 / 25

Reflection across the y-axis

A symmetry of a function can be represented by an algebra statement. Reflection across the y-axis interchanges positive x-values with negative x-values, swapping x and −x. Therefore f(−x) = f(x). The statement, “For all x ∈ R, f(−x) = f(x)” is equivalent to the statement “The graph of the function is unchanged by reflection across the y-axis.”

Smith (SHSU) Elementary Functions 2013 3 / 25

Rotation about the origin

What other symmetries might functions have? We can reflect a graph about the x-axis by replacing f(x) by −f(x). But could a graph be fixed by this reflection? Whenever a number is equal to its negative, then the number is zero. (x = −x = ⇒ 2x = 0 = ⇒ x = 0.) So if f(x) = −f(x) then f(x) = 0. But we could reflect a graph across first one axis and then the other. Reflecting a graph across the y-axis and then across the x-axis is equivalent to rotating the graph 180◦ around the origin. When this happens, f(x) = −f(−x). If f(x) = −f(−x) then we have rotational symmetry about the origin. In this case, we may multiply both sides of the equation by −1 and write f(−x) = −f(x).

Smith (SHSU) Elementary Functions 2013 4 / 25

Rotation about the origin

What other symmetries might functions have? We can reflect a graph about the x-axis by replacing f(x) by −f(x). But could a graph be fixed by this reflection? Whenever a number is equal to its negative, then the number is zero. (x = −x = ⇒ 2x = 0 = ⇒ x = 0.) So if f(x) = −f(x) then f(x) = 0. But we could reflect a graph across first one axis and then the other. Reflecting a graph across the y-axis and then across the x-axis is equivalent to rotating the graph 180◦ around the origin. When this happens, f(x) = −f(−x). If f(x) = −f(−x) then we have rotational symmetry about the origin. In this case, we may multiply both sides of the equation by −1 and write f(−x) = −f(x).

Smith (SHSU) Elementary Functions 2013 4 / 25

Rotation about the origin

What other symmetries might functions have? We can reflect a graph about the x-axis by replacing f(x) by −f(x). But could a graph be fixed by this reflection? Whenever a number is equal to its negative, then the number is zero. (x = −x = ⇒ 2x = 0 = ⇒ x = 0.) So if f(x) = −f(x) then f(x) = 0. But we could reflect a graph across first one axis and then the other. Reflecting a graph across the y-axis and then across the x-axis is equivalent to rotating the graph 180◦ around the origin. When this happens, f(x) = −f(−x). If f(x) = −f(−x) then we have rotational symmetry about the origin. In this case, we may multiply both sides of the equation by −1 and write f(−x) = −f(x).

Smith (SHSU) Elementary Functions 2013 4 / 25

Rotation about the origin

What other symmetries might functions have? We can reflect a graph about the x-axis by replacing f(x) by −f(x). But could a graph be fixed by this reflection? Whenever a number is equal to its negative, then the number is zero. (x = −x = ⇒ 2x = 0 = ⇒ x = 0.) So if f(x) = −f(x) then f(x) = 0. But we could reflect a graph across first one axis and then the other. Reflecting a graph across the y-axis and then across the x-axis is equivalent to rotating the graph 180◦ around the origin. When this happens, f(x) = −f(−x). If f(x) = −f(−x) then we have rotational symmetry about the origin. In this case, we may multiply both sides of the equation by −1 and write f(−x) = −f(x).

Smith (SHSU) Elementary Functions 2013 4 / 25

Rotation about the origin

What other symmetries might functions have? We can reflect a graph about the x-axis by replacing f(x) by −f(x). But could a graph be fixed by this reflection? Whenever a number is equal to its negative, then the number is zero. (x = −x = ⇒ 2x = 0 = ⇒ x = 0.) So if f(x) = −f(x) then f(x) = 0. But we could reflect a graph across first one axis and then the other. Reflecting a graph across the y-axis and then across the x-axis is equivalent to rotating the graph 180◦ around the origin. When this happens, f(x) = −f(−x). If f(x) = −f(−x) then we have rotational symmetry about the origin. In this case, we may multiply both sides of the equation by −1 and write f(−x) = −f(x).

Smith (SHSU) Elementary Functions 2013 4 / 25

Rotation about the origin

What other symmetries might functions have? We can reflect a graph about the x-axis by replacing f(x) by −f(x). But could a graph be fixed by this reflection? Whenever a number is equal to its negative, then the number is zero. (x = −x = ⇒ 2x = 0 = ⇒ x = 0.) So if f(x) = −f(x) then f(x) = 0. But we could reflect a graph across first one axis and then the other. Reflecting a graph across the y-axis and then across the x-axis is equivalent to rotating the graph 180◦ around the origin. When this happens, f(x) = −f(−x). If f(x) = −f(−x) then we have rotational symmetry about the origin. In this case, we may multiply both sides of the equation by −1 and write f(−x) = −f(x).

Smith (SHSU) Elementary Functions 2013 4 / 25

Rotation about the origin

What other symmetries might functions have? We can reflect a graph about the x-axis by replacing f(x) by −f(x). But could a graph be fixed by this reflection? Whenever a number is equal to its negative, then the number is zero. (x = −x = ⇒ 2x = 0 = ⇒ x = 0.) So if f(x) = −f(x) then f(x) = 0. But we could reflect a graph across first one axis and then the other. Reflecting a graph across the y-axis and then across the x-axis is equivalent to rotating the graph 180◦ around the origin. When this happens, f(x) = −f(−x). If f(x) = −f(−x) then we have rotational symmetry about the origin. In this case, we may multiply both sides of the equation by −1 and write f(−x) = −f(x).

Smith (SHSU) Elementary Functions 2013 4 / 25

Rotation about the origin

What other symmetries might functions have? We can reflect a graph about the x-axis by replacing f(x) by −f(x). But could a graph be fixed by this reflection? Whenever a number is equal to its negative, then the number is zero. (x = −x = ⇒ 2x = 0 = ⇒ x = 0.) So if f(x) = −f(x) then f(x) = 0. But we could reflect a graph across first one axis and then the other. Reflecting a graph across the y-axis and then across the x-axis is equivalent to rotating the graph 180◦ around the origin. When this happens, f(x) = −f(−x). If f(x) = −f(−x) then we have rotational symmetry about the origin. In this case, we may multiply both sides of the equation by −1 and write f(−x) = −f(x).

Smith (SHSU) Elementary Functions 2013 4 / 25

Rotation about the origin

What other symmetries might functions have? We can reflect a graph about the x-axis by replacing f(x) by −f(x). But could a graph be fixed by this reflection? Whenever a number is equal to its negative, then the number is zero. (x = −x = ⇒ 2x = 0 = ⇒ x = 0.) So if f(x) = −f(x) then f(x) = 0. But we could reflect a graph across first one axis and then the other. Reflecting a graph across the y-axis and then across the x-axis is equivalent to rotating the graph 180◦ around the origin. When this happens, f(x) = −f(−x). If f(x) = −f(−x) then we have rotational symmetry about the origin. In this case, we may multiply both sides of the equation by −1 and write f(−x) = −f(x).

Smith (SHSU) Elementary Functions 2013 4 / 25

Rotation about the origin

What other symmetries might functions have? We can reflect a graph about the x-axis by replacing f(x) by −f(x). But could a graph be fixed by this reflection? Whenever a number is equal to its negative, then the number is zero. (x = −x = ⇒ 2x = 0 = ⇒ x = 0.) So if f(x) = −f(x) then f(x) = 0. But we could reflect a graph across first one axis and then the other. Reflecting a graph across the y-axis and then across the x-axis is equivalent to rotating the graph 180◦ around the origin. When this happens, f(x) = −f(−x). If f(x) = −f(−x) then we have rotational symmetry about the origin. In this case, we may multiply both sides of the equation by −1 and write f(−x) = −f(x).

Smith (SHSU) Elementary Functions 2013 4 / 25

Rotation about the origin

What other symmetries might functions have? We can reflect a graph about the x-axis by replacing f(x) by −f(x). But could a graph be fixed by this reflection? Whenever a number is equal to its negative, then the number is zero. (x = −x = ⇒ 2x = 0 = ⇒ x = 0.) So if f(x) = −f(x) then f(x) = 0. But we could reflect a graph across first one axis and then the other. Reflecting a graph across the y-axis and then across the x-axis is equivalent to rotating the graph 180◦ around the origin. When this happens, f(x) = −f(−x). If f(x) = −f(−x) then we have rotational symmetry about the origin. In this case, we may multiply both sides of the equation by −1 and write f(−x) = −f(x).

Smith (SHSU) Elementary Functions 2013 4 / 25

Even and odd functions

So far, we have discussed two types of symmetry for graphs of functions:

1 Reflection symmetry about the y-axis, in which case f(−x) = f(x). 2 Rotation symmetry about the origin, in which case f(−x) = −f(x).

We note that functions like f(x) = x2 and f(x) = x4, where the exponent on x is even will have the property that f(−x) = f(x) since −1 to an even integer power is equal to 1. Similarly, functions like f(x) = x, f(x) = x3 and f(x) = x5, where the exponent

• n x is odd will have the property that f(−x) = −f(x) since −1 to an odd power

is equal to −1. This motivates the following definitions.

Smith (SHSU) Elementary Functions 2013 5 / 25

Even and odd functions

So far, we have discussed two types of symmetry for graphs of functions:

1 Reflection symmetry about the y-axis, in which case f(−x) = f(x). 2 Rotation symmetry about the origin, in which case f(−x) = −f(x).

We note that functions like f(x) = x2 and f(x) = x4, where the exponent on x is even will have the property that f(−x) = f(x) since −1 to an even integer power is equal to 1. Similarly, functions like f(x) = x, f(x) = x3 and f(x) = x5, where the exponent

• n x is odd will have the property that f(−x) = −f(x) since −1 to an odd power

is equal to −1. This motivates the following definitions.

Smith (SHSU) Elementary Functions 2013 5 / 25

Even and odd functions

So far, we have discussed two types of symmetry for graphs of functions:

1 Reflection symmetry about the y-axis, in which case f(−x) = f(x). 2 Rotation symmetry about the origin, in which case f(−x) = −f(x).

We note that functions like f(x) = x2 and f(x) = x4, where the exponent on x is even will have the property that f(−x) = f(x) since −1 to an even integer power is equal to 1. Similarly, functions like f(x) = x, f(x) = x3 and f(x) = x5, where the exponent

• n x is odd will have the property that f(−x) = −f(x) since −1 to an odd power

is equal to −1. This motivates the following definitions.

Smith (SHSU) Elementary Functions 2013 5 / 25

Even and odd functions

So far, we have discussed two types of symmetry for graphs of functions:

1 Reflection symmetry about the y-axis, in which case f(−x) = f(x). 2 Rotation symmetry about the origin, in which case f(−x) = −f(x).

We note that functions like f(x) = x2 and f(x) = x4, where the exponent on x is even will have the property that f(−x) = f(x) since −1 to an even integer power is equal to 1. Similarly, functions like f(x) = x, f(x) = x3 and f(x) = x5, where the exponent

• n x is odd will have the property that f(−x) = −f(x) since −1 to an odd power

is equal to −1. This motivates the following definitions.

Smith (SHSU) Elementary Functions 2013 5 / 25

Even and odd functions

So far, we have discussed two types of symmetry for graphs of functions:

1 Reflection symmetry about the y-axis, in which case f(−x) = f(x). 2 Rotation symmetry about the origin, in which case f(−x) = −f(x).

We note that functions like f(x) = x2 and f(x) = x4, where the exponent on x is even will have the property that f(−x) = f(x) since −1 to an even integer power is equal to 1. Similarly, functions like f(x) = x, f(x) = x3 and f(x) = x5, where the exponent

• n x is odd will have the property that f(−x) = −f(x) since −1 to an odd power

is equal to −1. This motivates the following definitions.

Smith (SHSU) Elementary Functions 2013 5 / 25

Even and odd functions

Definition. A function f(x) is even if f(−x) = f(x). The function is odd if f(−x) = −f(x). An even function has reflection symmetry about the y-axis. An odd function has rotational symmetry about the origin. We can decide algebraically if a function is even, odd or neither by replacing x by −x and computing f(−x). If f(−x) = f(x), the function is even. If f(−x) = −f(x), the function is odd.

Smith (SHSU) Elementary Functions 2013 6 / 25

Even and odd functions

Definition. A function f(x) is even if f(−x) = f(x). The function is odd if f(−x) = −f(x). An even function has reflection symmetry about the y-axis. An odd function has rotational symmetry about the origin. We can decide algebraically if a function is even, odd or neither by replacing x by −x and computing f(−x). If f(−x) = f(x), the function is even. If f(−x) = −f(x), the function is odd.

Smith (SHSU) Elementary Functions 2013 6 / 25

Even and odd functions

Definition. A function f(x) is even if f(−x) = f(x). The function is odd if f(−x) = −f(x). An even function has reflection symmetry about the y-axis. An odd function has rotational symmetry about the origin. We can decide algebraically if a function is even, odd or neither by replacing x by −x and computing f(−x). If f(−x) = f(x), the function is even. If f(−x) = −f(x), the function is odd.

Smith (SHSU) Elementary Functions 2013 6 / 25

Even and odd functions

Definition. A function f(x) is even if f(−x) = f(x). The function is odd if f(−x) = −f(x). An even function has reflection symmetry about the y-axis. An odd function has rotational symmetry about the origin. We can decide algebraically if a function is even, odd or neither by replacing x by −x and computing f(−x). If f(−x) = f(x), the function is even. If f(−x) = −f(x), the function is odd.

Smith (SHSU) Elementary Functions 2013 6 / 25

Even and odd functions

Definition. A function f(x) is even if f(−x) = f(x). The function is odd if f(−x) = −f(x). An even function has reflection symmetry about the y-axis. An odd function has rotational symmetry about the origin. We can decide algebraically if a function is even, odd or neither by replacing x by −x and computing f(−x). If f(−x) = f(x), the function is even. If f(−x) = −f(x), the function is odd.

Smith (SHSU) Elementary Functions 2013 6 / 25

Even and odd functions

Definition. A function f(x) is even if f(−x) = f(x). The function is odd if f(−x) = −f(x). An even function has reflection symmetry about the y-axis. An odd function has rotational symmetry about the origin. We can decide algebraically if a function is even, odd or neither by replacing x by −x and computing f(−x). If f(−x) = f(x), the function is even. If f(−x) = −f(x), the function is odd.

Smith (SHSU) Elementary Functions 2013 6 / 25

Even and odd functions

Definition. A function f(x) is even if f(−x) = f(x). The function is odd if f(−x) = −f(x). An even function has reflection symmetry about the y-axis. An odd function has rotational symmetry about the origin. We can decide algebraically if a function is even, odd or neither by replacing x by −x and computing f(−x). If f(−x) = f(x), the function is even. If f(−x) = −f(x), the function is odd.

Smith (SHSU) Elementary Functions 2013 6 / 25

Even and odd functions

• Examples. The graphs of a variety of functions are given below (on this page and

the next). Consider the symmetries of the graph y = f(x) and decide, from the graph drawings, if f(x) is odd, even or neither. Even Odd

Smith (SHSU) Elementary Functions 2013 7 / 25

Even and odd functions

• Examples. The graphs of a variety of functions are given below (on this page and

the next). Consider the symmetries of the graph y = f(x) and decide, from the graph drawings, if f(x) is odd, even or neither. Even Odd

Smith (SHSU) Elementary Functions 2013 7 / 25

Even and odd functions

• Examples. The graphs of a variety of functions are given below (on this page and

the next). Consider the symmetries of the graph y = f(x) and decide, from the graph drawings, if f(x) is odd, even or neither. Even Odd

Smith (SHSU) Elementary Functions 2013 7 / 25

Even and odd functions

Even Odd

Smith (SHSU) Elementary Functions 2013 8 / 25

Even and odd functions

Even Odd

Smith (SHSU) Elementary Functions 2013 8 / 25

Even and odd functions

Even Odd

Smith (SHSU) Elementary Functions 2013 8 / 25

Even and odd functions

Odd Odd

Smith (SHSU) Elementary Functions 2013 9 / 25

Even and odd functions

Odd Odd

Smith (SHSU) Elementary Functions 2013 9 / 25

Even and odd functions

Odd Odd

Smith (SHSU) Elementary Functions 2013 9 / 25

Even and odd functions

Even Odd

Smith (SHSU) Elementary Functions 2013 10 / 25

Even and odd functions

Even Odd

Smith (SHSU) Elementary Functions 2013 10 / 25

Even and odd functions

Even Odd

Smith (SHSU) Elementary Functions 2013 10 / 25

Even and odd functions

Odd Even

Smith (SHSU) Elementary Functions 2013 11 / 25

Even and odd functions

Odd Even

Smith (SHSU) Elementary Functions 2013 11 / 25

Even and odd functions

Odd Even

Smith (SHSU) Elementary Functions 2013 11 / 25

Even and odd functions, some examples

Three worked exercises.

1 Graph the function f(x) = x3 − 4x and then decide if the function is even,

• dd, or neither. Solution. This function is odd since it is symmetric about

the origin. We can check this algebraically: f(−x) = (−x)3 − 4(−x) = −x3 + 4x = −(x3 − 4x) = −f(x).

Smith (SHSU) Elementary Functions 2013 12 / 25

Even and odd functions, some examples

Three worked exercises.

1 Graph the function f(x) = x3 − 4x and then decide if the function is even,

• dd, or neither. Solution. This function is odd since it is symmetric about

the origin. We can check this algebraically: f(−x) = (−x)3 − 4(−x) = −x3 + 4x = −(x3 − 4x) = −f(x).

Smith (SHSU) Elementary Functions 2013 12 / 25

Even and odd functions, some examples

Three worked exercises.

1 Graph the function f(x) = x3 − 4x and then decide if the function is even,

• dd, or neither. Solution. This function is odd since it is symmetric about

the origin. We can check this algebraically: f(−x) = (−x)3 − 4(−x) = −x3 + 4x = −(x3 − 4x) = −f(x).

Smith (SHSU) Elementary Functions 2013 12 / 25

Even and odd functions, some examples

Three worked exercises.

1 Graph the function f(x) = x3 − 4x and then decide if the function is even,

• dd, or neither. Solution. This function is odd since it is symmetric about

the origin. We can check this algebraically: f(−x) = (−x)3 − 4(−x) = −x3 + 4x = −(x3 − 4x) = −f(x).

Smith (SHSU) Elementary Functions 2013 12 / 25

Even and odd functions, some examples

Three worked exercises.

1 Graph the function f(x) = x3 − 4x and then decide if the function is even,

• dd, or neither. Solution. This function is odd since it is symmetric about

the origin. We can check this algebraically: f(−x) = (−x)3 − 4(−x) = −x3 + 4x = −(x3 − 4x) = −f(x).

Smith (SHSU) Elementary Functions 2013 12 / 25

Even and odd functions, some examples

Three worked exercises.

1 Graph the function f(x) = x3 − 4x and then decide if the function is even,

• dd, or neither. Solution. This function is odd since it is symmetric about

the origin. We can check this algebraically: f(−x) = (−x)3 − 4(−x) = −x3 + 4x = −(x3 − 4x) = −f(x).

Smith (SHSU) Elementary Functions 2013 12 / 25

Even and odd functions, example 2

2 Decide algebraically if the function f(x) =

x 1 + x2 is even, odd, or neither. Solution. If f(x) = x 1 + x2 then f(−x) = −x 1 + (−x)2 . Since (−x)2 = x2 we can simplify this to f(−x) = −x 1 + (−x)2 = − x 1 + x2 = −f(x). So f(x) is odd.

Smith (SHSU) Elementary Functions 2013 13 / 25

Even and odd functions, example 2

2 Decide algebraically if the function f(x) =

x 1 + x2 is even, odd, or neither. Solution. If f(x) = x 1 + x2 then f(−x) = −x 1 + (−x)2 . Since (−x)2 = x2 we can simplify this to f(−x) = −x 1 + (−x)2 = − x 1 + x2 = −f(x). So f(x) is odd.

Smith (SHSU) Elementary Functions 2013 13 / 25

Even and odd functions, example 2

2 Decide algebraically if the function f(x) =

x 1 + x2 is even, odd, or neither. Solution. If f(x) = x 1 + x2 then f(−x) = −x 1 + (−x)2 . Since (−x)2 = x2 we can simplify this to f(−x) = −x 1 + (−x)2 = − x 1 + x2 = −f(x). So f(x) is odd.

Smith (SHSU) Elementary Functions 2013 13 / 25

Even and odd functions, example 2

2 Decide algebraically if the function f(x) =

x 1 + x2 is even, odd, or neither. Solution. If f(x) = x 1 + x2 then f(−x) = −x 1 + (−x)2 . Since (−x)2 = x2 we can simplify this to f(−x) = −x 1 + (−x)2 = − x 1 + x2 = −f(x). So f(x) is odd.

Smith (SHSU) Elementary Functions 2013 13 / 25

Even and odd functions, example 2

2 Decide algebraically if the function f(x) =

x 1 + x2 is even, odd, or neither. Solution. If f(x) = x 1 + x2 then f(−x) = −x 1 + (−x)2 . Since (−x)2 = x2 we can simplify this to f(−x) = −x 1 + (−x)2 = − x 1 + x2 = −f(x). So f(x) is odd.

Smith (SHSU) Elementary Functions 2013 13 / 25

Even and odd functions, example 2

2 Decide algebraically if the function f(x) =

x 1 + x2 is even, odd, or neither. Solution. If f(x) = x 1 + x2 then f(−x) = −x 1 + (−x)2 . Since (−x)2 = x2 we can simplify this to f(−x) = −x 1 + (−x)2 = − x 1 + x2 = −f(x). So f(x) is odd.

Smith (SHSU) Elementary Functions 2013 13 / 25

Even and odd functions, example 3

2 Decide algebraically if the function f(x) = x5 + 7x2 − 3x + 5 is even, odd, or

neither. Solution. If f(x) = x5 + 7x2 − 3x + 5 then f(−x) = (−x)5 + 7(−x)2 − 3(−x) + 5 = −x5 + 7x2 + 3x + 5. Since f(−x) = −x5 + 7x2 + 3x + 5 is neither equal to f(x) nor equal to −f(x) then f(x) is neither even nor odd.

Smith (SHSU) Elementary Functions 2013 14 / 25

Even and odd functions, example 3

2 Decide algebraically if the function f(x) = x5 + 7x2 − 3x + 5 is even, odd, or

neither. Solution. If f(x) = x5 + 7x2 − 3x + 5 then f(−x) = (−x)5 + 7(−x)2 − 3(−x) + 5 = −x5 + 7x2 + 3x + 5. Since f(−x) = −x5 + 7x2 + 3x + 5 is neither equal to f(x) nor equal to −f(x) then f(x) is neither even nor odd.

Smith (SHSU) Elementary Functions 2013 14 / 25

Even and odd functions, example 3

2 Decide algebraically if the function f(x) = x5 + 7x2 − 3x + 5 is even, odd, or

neither. Solution. If f(x) = x5 + 7x2 − 3x + 5 then f(−x) = (−x)5 + 7(−x)2 − 3(−x) + 5 = −x5 + 7x2 + 3x + 5. Since f(−x) = −x5 + 7x2 + 3x + 5 is neither equal to f(x) nor equal to −f(x) then f(x) is neither even nor odd.

Smith (SHSU) Elementary Functions 2013 14 / 25

Even and odd functions, example 3

2 Decide algebraically if the function f(x) = x5 + 7x2 − 3x + 5 is even, odd, or

neither. Solution. If f(x) = x5 + 7x2 − 3x + 5 then f(−x) = (−x)5 + 7(−x)2 − 3(−x) + 5 = −x5 + 7x2 + 3x + 5. Since f(−x) = −x5 + 7x2 + 3x + 5 is neither equal to f(x) nor equal to −f(x) then f(x) is neither even nor odd.

Smith (SHSU) Elementary Functions 2013 14 / 25

Even and odd functions, example 3

2 Decide algebraically if the function f(x) = x5 + 7x2 − 3x + 5 is even, odd, or

neither. Solution. If f(x) = x5 + 7x2 − 3x + 5 then f(−x) = (−x)5 + 7(−x)2 − 3(−x) + 5 = −x5 + 7x2 + 3x + 5. Since f(−x) = −x5 + 7x2 + 3x + 5 is neither equal to f(x) nor equal to −f(x) then f(x) is neither even nor odd.

Smith (SHSU) Elementary Functions 2013 14 / 25

Even and odd functions, example 3

2 Decide algebraically if the function f(x) = x5 + 7x2 − 3x + 5 is even, odd, or

neither. Solution. If f(x) = x5 + 7x2 − 3x + 5 then f(−x) = (−x)5 + 7(−x)2 − 3(−x) + 5 = −x5 + 7x2 + 3x + 5. Since f(−x) = −x5 + 7x2 + 3x + 5 is neither equal to f(x) nor equal to −f(x) then f(x) is neither even nor odd.

Smith (SHSU) Elementary Functions 2013 14 / 25

Even and odd functions: can a function be both??

Testing the concepts. There is a function which is both even and odd! What is it?

??

(END)

Smith (SHSU) Elementary Functions 2013 15 / 25

Even and odd functions: can a function be both??

Testing the concepts. There is a function which is both even and odd! What is it?

??

(END)

Smith (SHSU) Elementary Functions 2013 15 / 25

Even and odd functions: can a function be both??

Testing the concepts. There is a function which is both even and odd! What is it?

??

(END)

Smith (SHSU) Elementary Functions 2013 15 / 25