Elementary Functions Part 1, Functions Lecture 1.1c, Finding the - - PowerPoint PPT Presentation

Elementary Functions

Part 1, Functions Lecture 1.1c, Finding the domains of functions

• Dr. Ken W. Smith

Sam Houston State University

2013

Smith (SHSU) Elementary Functions 2013 22 / 27

Domains of functions

The domain of a function is (generally) the largest possible set of inputs into the

• function. Let’s find the domain of the function

f(x) = √x. It is often easier to ask the question, “What is not in the domain?”. For the function f(x) = √x we ask the question, “Which real numbers do not have a square root?” We cannot evaluate f(x) at negative numbers since the square of a real number cannot be negative. So the domain must be numbers which are not negative, that is, zero and positive real numbers. (We can indeed take the square root of 0 so we want to include 0 in the domain.) We can write our answer in interval notation:

• Solution. The domain of f(x) = √x is

[0, ∞).

Smith (SHSU) Elementary Functions 2013 23 / 27

Domains of functions

The domain of a function is (generally) the largest possible set of inputs into the

• function. Let’s find the domain of the function

f(x) = √x. It is often easier to ask the question, “What is not in the domain?”. For the function f(x) = √x we ask the question, “Which real numbers do not have a square root?” We cannot evaluate f(x) at negative numbers since the square of a real number cannot be negative. So the domain must be numbers which are not negative, that is, zero and positive real numbers. (We can indeed take the square root of 0 so we want to include 0 in the domain.) We can write our answer in interval notation:

• Solution. The domain of f(x) = √x is

[0, ∞).

Smith (SHSU) Elementary Functions 2013 23 / 27

Domains of functions

The domain of a function is (generally) the largest possible set of inputs into the

• function. Let’s find the domain of the function

f(x) = √x. It is often easier to ask the question, “What is not in the domain?”. For the function f(x) = √x we ask the question, “Which real numbers do not have a square root?” We cannot evaluate f(x) at negative numbers since the square of a real number cannot be negative. So the domain must be numbers which are not negative, that is, zero and positive real numbers. (We can indeed take the square root of 0 so we want to include 0 in the domain.) We can write our answer in interval notation:

• Solution. The domain of f(x) = √x is

[0, ∞).

Smith (SHSU) Elementary Functions 2013 23 / 27

Domains of functions

The domain of a function is (generally) the largest possible set of inputs into the

• function. Let’s find the domain of the function

f(x) = √x. It is often easier to ask the question, “What is not in the domain?”. For the function f(x) = √x we ask the question, “Which real numbers do not have a square root?” We cannot evaluate f(x) at negative numbers since the square of a real number cannot be negative. So the domain must be numbers which are not negative, that is, zero and positive real numbers. (We can indeed take the square root of 0 so we want to include 0 in the domain.) We can write our answer in interval notation:

• Solution. The domain of f(x) = √x is

[0, ∞).

Smith (SHSU) Elementary Functions 2013 23 / 27

Domains of functions

The domain of a function is (generally) the largest possible set of inputs into the

• function. Let’s find the domain of the function

f(x) = √x. It is often easier to ask the question, “What is not in the domain?”. For the function f(x) = √x we ask the question, “Which real numbers do not have a square root?” We cannot evaluate f(x) at negative numbers since the square of a real number cannot be negative. So the domain must be numbers which are not negative, that is, zero and positive real numbers. (We can indeed take the square root of 0 so we want to include 0 in the domain.) We can write our answer in interval notation:

• Solution. The domain of f(x) = √x is

[0, ∞).

Smith (SHSU) Elementary Functions 2013 23 / 27

Domains of functions

The domain of a function is (generally) the largest possible set of inputs into the

• function. Let’s find the domain of the function

f(x) = √x. It is often easier to ask the question, “What is not in the domain?”. For the function f(x) = √x we ask the question, “Which real numbers do not have a square root?” We cannot evaluate f(x) at negative numbers since the square of a real number cannot be negative. So the domain must be numbers which are not negative, that is, zero and positive real numbers. (We can indeed take the square root of 0 so we want to include 0 in the domain.) We can write our answer in interval notation:

• Solution. The domain of f(x) = √x is

[0, ∞).

Smith (SHSU) Elementary Functions 2013 23 / 27

Domains of functions

The domain of a function is (generally) the largest possible set of inputs into the

• function. Let’s find the domain of the function

f(x) = √x. It is often easier to ask the question, “What is not in the domain?”. For the function f(x) = √x we ask the question, “Which real numbers do not have a square root?” We cannot evaluate f(x) at negative numbers since the square of a real number cannot be negative. So the domain must be numbers which are not negative, that is, zero and positive real numbers. (We can indeed take the square root of 0 so we want to include 0 in the domain.) We can write our answer in interval notation:

• Solution. The domain of f(x) = √x is

[0, ∞).

Smith (SHSU) Elementary Functions 2013 23 / 27

Domains of functions

The domain of a function is (generally) the largest possible set of inputs into the

• function. Let’s find the domain of the function

f(x) = √x. It is often easier to ask the question, “What is not in the domain?”. For the function f(x) = √x we ask the question, “Which real numbers do not have a square root?” We cannot evaluate f(x) at negative numbers since the square of a real number cannot be negative. So the domain must be numbers which are not negative, that is, zero and positive real numbers. (We can indeed take the square root of 0 so we want to include 0 in the domain.) We can write our answer in interval notation:

• Solution. The domain of f(x) = √x is

[0, ∞).

Smith (SHSU) Elementary Functions 2013 23 / 27

Domains of functions

The domain of a function is (generally) the largest possible set of inputs into the

• function. Let’s find the domain of the function

f(x) = √x. It is often easier to ask the question, “What is not in the domain?”. For the function f(x) = √x we ask the question, “Which real numbers do not have a square root?” We cannot evaluate f(x) at negative numbers since the square of a real number cannot be negative. So the domain must be numbers which are not negative, that is, zero and positive real numbers. (We can indeed take the square root of 0 so we want to include 0 in the domain.) We can write our answer in interval notation:

• Solution. The domain of f(x) = √x is

[0, ∞).

Smith (SHSU) Elementary Functions 2013 23 / 27

Definition of a function

• Example. Find the domain of the function g(x) =

1 x + 2 + 2x − 3 2x + 1 + x − 5.

• Solution. What numbers cannot serve as input to g(x)? Since we cannot have

denominators equal to zero then x = −2 cannot be an input; neither can x = − 1

2.

So the domain of this function g is all real numbers except x = −2 and x = − 1

2.

There are several ways to write the domain of g. Using set notation, we could write the domain as {x ∈ R : x = −2, −1 2}. This is a precise symbolic way to say, “All real numbers except −2 and − 1

2.”

We could also write the domain in interval notation: (−∞, −2) ∪ (−2, −1 2) ∪ (−1 2, ∞). This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 1

2, along with (in addition) the

real numbers larger than − 1

2.

Smith (SHSU) Elementary Functions 2013 24 / 27

Definition of a function

• Example. Find the domain of the function g(x) =

1 x + 2 + 2x − 3 2x + 1 + x − 5.

• Solution. What numbers cannot serve as input to g(x)? Since we cannot have

denominators equal to zero then x = −2 cannot be an input; neither can x = − 1

2.

So the domain of this function g is all real numbers except x = −2 and x = − 1

2.

There are several ways to write the domain of g. Using set notation, we could write the domain as {x ∈ R : x = −2, −1 2}. This is a precise symbolic way to say, “All real numbers except −2 and − 1

2.”

We could also write the domain in interval notation: (−∞, −2) ∪ (−2, −1 2) ∪ (−1 2, ∞). This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 1

2, along with (in addition) the

real numbers larger than − 1

2.

Smith (SHSU) Elementary Functions 2013 24 / 27

Definition of a function

• Example. Find the domain of the function g(x) =

1 x + 2 + 2x − 3 2x + 1 + x − 5.

• Solution. What numbers cannot serve as input to g(x)? Since we cannot have

denominators equal to zero then x = −2 cannot be an input; neither can x = − 1

2.

So the domain of this function g is all real numbers except x = −2 and x = − 1

2.

There are several ways to write the domain of g. Using set notation, we could write the domain as {x ∈ R : x = −2, −1 2}. This is a precise symbolic way to say, “All real numbers except −2 and − 1

2.”

We could also write the domain in interval notation: (−∞, −2) ∪ (−2, −1 2) ∪ (−1 2, ∞). This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 1

2, along with (in addition) the

real numbers larger than − 1

2.

Smith (SHSU) Elementary Functions 2013 24 / 27

Definition of a function

• Example. Find the domain of the function g(x) =

1 x + 2 + 2x − 3 2x + 1 + x − 5.

• Solution. What numbers cannot serve as input to g(x)? Since we cannot have

denominators equal to zero then x = −2 cannot be an input; neither can x = − 1

2.

So the domain of this function g is all real numbers except x = −2 and x = − 1

2.

There are several ways to write the domain of g. Using set notation, we could write the domain as {x ∈ R : x = −2, −1 2}. This is a precise symbolic way to say, “All real numbers except −2 and − 1

2.”

We could also write the domain in interval notation: (−∞, −2) ∪ (−2, −1 2) ∪ (−1 2, ∞). This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 1

2, along with (in addition) the

real numbers larger than − 1

2.

Smith (SHSU) Elementary Functions 2013 24 / 27

Definition of a function

• Example. Find the domain of the function g(x) =

1 x + 2 + 2x − 3 2x + 1 + x − 5.

• Solution. What numbers cannot serve as input to g(x)? Since we cannot have

denominators equal to zero then x = −2 cannot be an input; neither can x = − 1

2.

So the domain of this function g is all real numbers except x = −2 and x = − 1

2.

There are several ways to write the domain of g. Using set notation, we could write the domain as {x ∈ R : x = −2, −1 2}. This is a precise symbolic way to say, “All real numbers except −2 and − 1

2.”

We could also write the domain in interval notation: (−∞, −2) ∪ (−2, −1 2) ∪ (−1 2, ∞). This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 1

2, along with (in addition) the

real numbers larger than − 1

2.

Smith (SHSU) Elementary Functions 2013 24 / 27

Definition of a function

• Example. Find the domain of the function g(x) =

1 x + 2 + 2x − 3 2x + 1 + x − 5.

• Solution. What numbers cannot serve as input to g(x)? Since we cannot have

denominators equal to zero then x = −2 cannot be an input; neither can x = − 1

2.

So the domain of this function g is all real numbers except x = −2 and x = − 1

2.

There are several ways to write the domain of g. Using set notation, we could write the domain as {x ∈ R : x = −2, −1 2}. This is a precise symbolic way to say, “All real numbers except −2 and − 1

2.”

We could also write the domain in interval notation: (−∞, −2) ∪ (−2, −1 2) ∪ (−1 2, ∞). This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 1

2, along with (in addition) the

real numbers larger than − 1

2.

Smith (SHSU) Elementary Functions 2013 24 / 27

Definition of a function

• Example. Find the domain of the function g(x) =

1 x + 2 + 2x − 3 2x + 1 + x − 5.

• Solution. What numbers cannot serve as input to g(x)? Since we cannot have

denominators equal to zero then x = −2 cannot be an input; neither can x = − 1

2.

So the domain of this function g is all real numbers except x = −2 and x = − 1

2.

There are several ways to write the domain of g. Using set notation, we could write the domain as {x ∈ R : x = −2, −1 2}. This is a precise symbolic way to say, “All real numbers except −2 and − 1

2.”

We could also write the domain in interval notation: (−∞, −2) ∪ (−2, −1 2) ∪ (−1 2, ∞). This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 1

2, along with (in addition) the

real numbers larger than − 1

2.

Smith (SHSU) Elementary Functions 2013 24 / 27

Definition of a function

• Example. Find the domain of the function g(x) =

1 x + 2 + 2x − 3 2x + 1 + x − 5.

• Solution. What numbers cannot serve as input to g(x)? Since we cannot have

denominators equal to zero then x = −2 cannot be an input; neither can x = − 1

2.

So the domain of this function g is all real numbers except x = −2 and x = − 1

2.

There are several ways to write the domain of g. Using set notation, we could write the domain as {x ∈ R : x = −2, −1 2}. This is a precise symbolic way to say, “All real numbers except −2 and − 1

2.”

We could also write the domain in interval notation: (−∞, −2) ∪ (−2, −1 2) ∪ (−1 2, ∞). This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 1

2, along with (in addition) the

real numbers larger than − 1

2.

Smith (SHSU) Elementary Functions 2013 24 / 27

Definition of a function

• Example. Find the domain of the function g(x) =

1 x + 2 + 2x − 3 2x + 1 + x − 5.

• Solution. What numbers cannot serve as input to g(x)? Since we cannot have

denominators equal to zero then x = −2 cannot be an input; neither can x = − 1

2.

So the domain of this function g is all real numbers except x = −2 and x = − 1

2.

There are several ways to write the domain of g. Using set notation, we could write the domain as {x ∈ R : x = −2, −1 2}. This is a precise symbolic way to say, “All real numbers except −2 and − 1

2.”

We could also write the domain in interval notation: (−∞, −2) ∪ (−2, −1 2) ∪ (−1 2, ∞). This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 1

2, along with (in addition) the

real numbers larger than − 1

2.

Smith (SHSU) Elementary Functions 2013 24 / 27

Definition of a function

• Example. Find the domain of the function g(x) =

1 x + 2 + 2x − 3 2x + 1 + x − 5.

• Solution. What numbers cannot serve as input to g(x)? Since we cannot have

denominators equal to zero then x = −2 cannot be an input; neither can x = − 1

2.

So the domain of this function g is all real numbers except x = −2 and x = − 1

2.

There are several ways to write the domain of g. Using set notation, we could write the domain as {x ∈ R : x = −2, −1 2}. This is a precise symbolic way to say, “All real numbers except −2 and − 1

2.”

We could also write the domain in interval notation: (−∞, −2) ∪ (−2, −1 2) ∪ (−1 2, ∞). This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 1

2, along with (in addition) the

real numbers larger than − 1

2.

Smith (SHSU) Elementary Functions 2013 24 / 27

Definition of a function

• Example. Find the domain of the function g(x) =

1 x + 2 + 2x − 3 2x + 1 + x − 5.

• Solution. What numbers cannot serve as input to g(x)? Since we cannot have

denominators equal to zero then x = −2 cannot be an input; neither can x = − 1

2.

So the domain of this function g is all real numbers except x = −2 and x = − 1

2.

There are several ways to write the domain of g. Using set notation, we could write the domain as {x ∈ R : x = −2, −1 2}. This is a precise symbolic way to say, “All real numbers except −2 and − 1

2.”

We could also write the domain in interval notation: (−∞, −2) ∪ (−2, −1 2) ∪ (−1 2, ∞). This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 1

2, along with (in addition) the

real numbers larger than − 1

2.

Smith (SHSU) Elementary Functions 2013 24 / 27

Definition of a function

Some worked exercises.

1 Find the domain of the function f(x) =

√ x − 1 Solution. Since the square root function requires nonnegative inputs, we must have x − 1 ≥ 0. Therefore we must have x ≥ 1. The domain is [1, ∞).

Smith (SHSU) Elementary Functions 2013 25 / 27

Definition of a function

Some worked exercises.

1 Find the domain of the function f(x) =

√ x − 1 Solution. Since the square root function requires nonnegative inputs, we must have x − 1 ≥ 0. Therefore we must have x ≥ 1. The domain is [1, ∞).

Smith (SHSU) Elementary Functions 2013 25 / 27

Definition of a function

Some worked exercises.

1 Find the domain of the function f(x) =

√ x − 1 Solution. Since the square root function requires nonnegative inputs, we must have x − 1 ≥ 0. Therefore we must have x ≥ 1. The domain is [1, ∞).

Smith (SHSU) Elementary Functions 2013 25 / 27

Definition of a function

Some worked exercises.

1 Find the domain of the function f(x) =

√ x − 1 Solution. Since the square root function requires nonnegative inputs, we must have x − 1 ≥ 0. Therefore we must have x ≥ 1. The domain is [1, ∞).

Smith (SHSU) Elementary Functions 2013 25 / 27

Definition of a function

2 Find the domain of the function f(x) =

√x − 1 x − 3

• Solution. Again, we must have x ≥ 1 but we must also prevent the

denominator from being zero, so x cannot be 3, either. The domain is then all real numbers at least as big as 1 except for the number 3. Here is our answer in interval notation: The domain is [1, 3) ∪ (3, ∞).

Smith (SHSU) Elementary Functions 2013 26 / 27

Definition of a function

2 Find the domain of the function f(x) =

√x − 1 x − 3

• Solution. Again, we must have x ≥ 1 but we must also prevent the

denominator from being zero, so x cannot be 3, either. The domain is then all real numbers at least as big as 1 except for the number 3. Here is our answer in interval notation: The domain is [1, 3) ∪ (3, ∞).

Smith (SHSU) Elementary Functions 2013 26 / 27

Definition of a function

2 Find the domain of the function f(x) =

√x − 1 x − 3

• Solution. Again, we must have x ≥ 1 but we must also prevent the

denominator from being zero, so x cannot be 3, either. The domain is then all real numbers at least as big as 1 except for the number 3. Here is our answer in interval notation: The domain is [1, 3) ∪ (3, ∞).

Smith (SHSU) Elementary Functions 2013 26 / 27

Definition of a function

2 Find the domain of the function f(x) =

√x − 1 x − 3

• Solution. Again, we must have x ≥ 1 but we must also prevent the

denominator from being zero, so x cannot be 3, either. The domain is then all real numbers at least as big as 1 except for the number 3. Here is our answer in interval notation: The domain is [1, 3) ∪ (3, ∞).

Smith (SHSU) Elementary Functions 2013 26 / 27

Definition of a function

2 Find the domain of the function f(x) =

√x − 1 x − 3

• Solution. Again, we must have x ≥ 1 but we must also prevent the

denominator from being zero, so x cannot be 3, either. The domain is then all real numbers at least as big as 1 except for the number 3. Here is our answer in interval notation: The domain is [1, 3) ∪ (3, ∞).

Smith (SHSU) Elementary Functions 2013 26 / 27

Definition of a function

3 Find the domain of the function f(x) =

√x − 1 x2 − 6x + 8

• Solution. We must have x ≥ 1 and we must prevent the denominator from

being zero. The denominator factors as x2 − 6x + 8 = (x − 2)(x − 4), so x cannot be 2 or 4. So our answer is all real numbers at least as big as 1 and not equal to 2 or 4. In interval notation, our answer is: The domain is [1, 2) ∪ (2, 4) ∪ (4, ∞.) (END)

Smith (SHSU) Elementary Functions 2013 27 / 27

Definition of a function

3 Find the domain of the function f(x) =

√x − 1 x2 − 6x + 8

• Solution. We must have x ≥ 1 and we must prevent the denominator from

being zero. The denominator factors as x2 − 6x + 8 = (x − 2)(x − 4), so x cannot be 2 or 4. So our answer is all real numbers at least as big as 1 and not equal to 2 or 4. In interval notation, our answer is: The domain is [1, 2) ∪ (2, 4) ∪ (4, ∞.) (END)

Smith (SHSU) Elementary Functions 2013 27 / 27

Definition of a function

3 Find the domain of the function f(x) =

√x − 1 x2 − 6x + 8

• Solution. We must have x ≥ 1 and we must prevent the denominator from

being zero. The denominator factors as x2 − 6x + 8 = (x − 2)(x − 4), so x cannot be 2 or 4. So our answer is all real numbers at least as big as 1 and not equal to 2 or 4. In interval notation, our answer is: The domain is [1, 2) ∪ (2, 4) ∪ (4, ∞.) (END)

Smith (SHSU) Elementary Functions 2013 27 / 27

Definition of a function

3 Find the domain of the function f(x) =

√x − 1 x2 − 6x + 8

• Solution. We must have x ≥ 1 and we must prevent the denominator from

being zero. The denominator factors as x2 − 6x + 8 = (x − 2)(x − 4), so x cannot be 2 or 4. So our answer is all real numbers at least as big as 1 and not equal to 2 or 4. In interval notation, our answer is: The domain is [1, 2) ∪ (2, 4) ∪ (4, ∞.) (END)

Smith (SHSU) Elementary Functions 2013 27 / 27

Definition of a function

3 Find the domain of the function f(x) =

√x − 1 x2 − 6x + 8

• Solution. We must have x ≥ 1 and we must prevent the denominator from

being zero. The denominator factors as x2 − 6x + 8 = (x − 2)(x − 4), so x cannot be 2 or 4. So our answer is all real numbers at least as big as 1 and not equal to 2 or 4. In interval notation, our answer is: The domain is [1, 2) ∪ (2, 4) ∪ (4, ∞.) (END)

Smith (SHSU) Elementary Functions 2013 27 / 27

Definition of a function

3 Find the domain of the function f(x) =

√x − 1 x2 − 6x + 8

• Solution. We must have x ≥ 1 and we must prevent the denominator from

being zero. The denominator factors as x2 − 6x + 8 = (x − 2)(x − 4), so x cannot be 2 or 4. So our answer is all real numbers at least as big as 1 and not equal to 2 or 4. In interval notation, our answer is: The domain is [1, 2) ∪ (2, 4) ∪ (4, ∞.) (END)

Smith (SHSU) Elementary Functions 2013 27 / 27

Definition of a function

3 Find the domain of the function f(x) =

√x − 1 x2 − 6x + 8

• Solution. We must have x ≥ 1 and we must prevent the denominator from

being zero. The denominator factors as x2 − 6x + 8 = (x − 2)(x − 4), so x cannot be 2 or 4. So our answer is all real numbers at least as big as 1 and not equal to 2 or 4. In interval notation, our answer is: The domain is [1, 2) ∪ (2, 4) ∪ (4, ∞.) (END)

Smith (SHSU) Elementary Functions 2013 27 / 27