Sections 3.1 - 3.3 Dr. Doug Ensley January 13, 2015 Function - - PowerPoint PPT Presentation

Sections 3.1 - 3.3

• Dr. Doug Ensley

January 13, 2015

Function Machine

◮ A function accepts an input x and returns an output f (x). ◮ The set of all legitimate inputs is called the domain, and the

set of all outputs that occur is called the range.

◮ A function can associate only one output to any given input.

It is ok for a function to have the same output for more than

• ne input.

Function Example

◮ The function with the rule, “For each input number x, return

as output the number x2 + 1,” can be written more concisely as, f (x) = x2 + 1. When we can provide such an algebraic rule, then the definition of function is usually met.

Function Example

◮ The function with the rule, “For each input number x, return

as output the number x2 + 1,” can be written more concisely as, f (x) = x2 + 1. When we can provide such an algebraic rule, then the definition of function is usually met.

◮ The function f (x) = x2 + 1 has as its domain the set of all

real numbers, denoted R, or in interval notation (−∞, ∞).

Function Example

◮ The function with the rule, “For each input number x, return

as output the number x2 + 1,” can be written more concisely as, f (x) = x2 + 1. When we can provide such an algebraic rule, then the definition of function is usually met.

◮ The function f (x) = x2 + 1 has as its domain the set of all

real numbers, denoted R, or in interval notation (−∞, ∞).

◮ The graph of a function consists of all points (x, y) where y

is the output associated with input x. I will refer to such points as “input, output pairs,” and we often use the notation y = f (x) when referring to a graph of a function.

Function Evaluation

Let f (x) = x2 − 2x + 3. Evaluate each of the following:

◮ f (−3)

Function Evaluation

Let f (x) = x2 − 2x + 3. Evaluate each of the following:

◮ f (−3) ◮ f

1

2

Function Evaluation

Let f (x) = x2 − 2x + 3. Evaluate each of the following:

◮ f (−3) ◮ f

1

2

• ◮ f (−x) (and simplify)

Function Evaluation

Let f (x) = x2 − 2x + 3. Evaluate each of the following:

◮ f (−3) ◮ f

1

2

• ◮ f (−x) (and simplify)

◮ f (x + h) (and simplify)

Function Evaluation

Let f (x) = x2 − 2x + 3. Evaluate each of the following:

◮ f (−3) ◮ f

1

2

• ◮ f (−x) (and simplify)

◮ f (x + h) (and simplify) ◮ f (x + h) − f (x)

h (and simplify)

Function Evaluation

Let f (x) = x2 + 1. Evaluate each of the following:

◮ f (7)

Function Evaluation

Let f (x) = x2 + 1. Evaluate each of the following:

◮ f (7) ◮ f

1

3

Function Evaluation

Let f (x) = x2 + 1. Evaluate each of the following:

◮ f (7) ◮ f

1

3

• ◮ f (2x) (and simplify)

Function Evaluation

Let f (x) = x2 + 1. Evaluate each of the following:

◮ f (7) ◮ f

1

3

• ◮ f (2x) (and simplify)

◮ f (x + h) (and simplify)

Function Evaluation

Let f (x) = x2 + 1. Evaluate each of the following:

◮ f (7) ◮ f

1

3

• ◮ f (2x) (and simplify)

◮ f (x + h) (and simplify) ◮ f (x + h) − f (x)

h (and simplify)

More Function Examples

◮ The function with the rule, “For each input number x, return

as output the number

x+1 √x−3,” can be written more concisely

as, g(x) = x + 1 √x − 3

More Function Examples

◮ The function with the rule, “For each input number x, return

as output the number

x+1 √x−3,” can be written more concisely

as, g(x) = x + 1 √x − 3

◮ The function g(x) has as its domain the set of all real

numbers greater than 3.

More Function Examples

◮ The function with the rule, “For each input number x, return

as output the number

x+1 √x−3,” can be written more concisely

as, g(x) = x + 1 √x − 3

◮ The function g(x) has as its domain the set of all real

numbers greater than 3.

◮ In interval notation, the domain is written (3, ∞).

Domain

◮ What is the domain of the function g(x) = √2x − 5?

Domain

◮ What is the domain of the function g(x) = √2x − 5? ◮ What is the domain of the function g(x) = x+1 x−4?

Domain

◮ What is the domain of the function g(x) = √2x − 5? ◮ What is the domain of the function g(x) = x+1 x−4? ◮ What is the domain of the function g(x) = x2+x+1 √3x−6 ?

Domain

◮ What is the domain of the function g(x) = √2x − 5? ◮ What is the domain of the function g(x) = x+1 x−4? ◮ What is the domain of the function g(x) = x2+x+1 √3x−6 ? ◮ What is the domain of the function g(x) = 1 x2+5?

Functions from Word Problems

◮ The function with the rule, “For each input number x, return

as output the price in dollars of purchasing x candy bars that cost $0.79 each,” can be written algebraically as, P(x) = 0.79 · x. What is the domain of the function P(x)?

Functions from Word Problems

◮ The function with the rule, “For each input number x, return

as output the price in dollars of purchasing x candy bars that cost $0.79 each,” can be written algebraically as, P(x) = 0.79 · x. What is the domain of the function P(x)?

◮ A rectangular garden has a perimeter of 100 feet. Find an

algebraic expression for the area A(w) of the garden if the width is the input number w. What is the domain of the function A(w)?

Functions from Word Problems

◮ The function with the rule, “For each input number x, return

as output the price in dollars of purchasing x candy bars that cost $0.79 each,” can be written algebraically as, P(x) = 0.79 · x. What is the domain of the function P(x)?

◮ A rectangular garden has a perimeter of 100 feet. Find an

algebraic expression for the area A(w) of the garden if the width is the input number w. What is the domain of the function A(w)?

◮ A popular burger restaurant gets a revenue R, in dollars, from

the sale of x hundred burgers equal to R(x) = −2.4x2 + 440x. The cost C, in dollars, of selling x hundred burgers is given by the function C(x) = 0.1x3 − 4x2 + 130x + 1000. Find the profit if 7 hundred burgers are sold. If P(x) denotes the profit when selling x hundred burgers, find P(7) and write a sentence explaining its meaning.

Graph of a function

◮ The graph of a function consists of all points (x, y) where y

is the output associated with input x. I will refer to such points as “input, output pairs,” and we often use the notation y = f (x) when referring to a graph of a function.

Graph of a function

◮ The graph of a function consists of all points (x, y) where y

is the output associated with input x. I will refer to such points as “input, output pairs,” and we often use the notation y = f (x) when referring to a graph of a function.

◮ For example, the graph of f (x) = 1 4x2 + 1 is shown below.

Graph of a function

◮ The graph of a function consists of all points (x, y) where y

is the output associated with input x. I will refer to such points as “input, output pairs,” and we often use the notation y = f (x) when referring to a graph of a function.

◮ For example, the graph of f (x) = 1 4x2 + 1 is shown below. ◮ When we want to refer to a point on the graph, we can

specify just the x coordinate or we can describe both coordinates as, for example, (−2, f (−2)).

Graph of a function

Consider the graph y = f (x) shown below:

◮ Compute f (3) and f (6).

Graph of a function

Consider the graph y = f (x) shown below:

◮ Compute f (3) and f (6). ◮ The point (8, a) is on the graph. What is a? The point

(b, −2) is on the graph. What is b?

Graph of a function

Consider the graph y = f (x) shown below:

◮ Compute f (3) and f (6). ◮ The point (8, a) is on the graph. What is a? The point

(b, −2) is on the graph. What is b?

◮ How many solutions does the equation f(x)=1 have?

Graph of a function

Consider the graph y = f (x) shown below:

◮ Compute f (3) and f (6). ◮ The point (8, a) is on the graph. What is a? The point

(b, −2) is on the graph. What is b?

◮ How many solutions does the equation f(x)=1 have? ◮ Identify the x-intercepts for this function?

Graph of a function

Consider the graph y = f (x) shown below:

◮ Compute f (3) and f (6). ◮ The point (8, a) is on the graph. What is a? The point

(b, −2) is on the graph. What is b?

◮ How many solutions does the equation f(x)=1 have? ◮ Identify the x-intercepts for this function? ◮ Identify the y-intercepts for this function?

Graph of a function

Consider the graph y = f (x) shown below:

◮ Compute f (3) and f (6). ◮ The point (8, a) is on the graph. What is a? The point

(b, −2) is on the graph. What is b?

◮ How many solutions does the equation f(x)=1 have? ◮ Identify the x-intercepts for this function? ◮ Identify the y-intercepts for this function? ◮ What is the domain of this function? What is the range?

Increasing/Decreasing

Consider the following graph of y = f (x).

◮ Where is f increasing? ◮ Where is f decreasing? ◮ Where are the “turning

points?”

Local extrema

Where are the local maxima? What are the local maxima? Are there global extrema?

Average Rate of Change

To get to Pittsburgh last week, I got on the turnpike at mile marker 202 at 1:00 and got off at mile marker 57 at 3:15. What was my average speed on the turnpike?

Average Rate of Change

To get to Pittsburgh last week, I got on the turnpike at mile marker 202 at 1:00 and got off at mile marker 57 at 3:15. What was my average speed on the turnpike? For a function f , the average rate of change of f over the interval [a, b] is given by f (b) − f (a) b − a

Average Rate of Change

To get to Pittsburgh last week, I got on the turnpike at mile marker 202 at 1:00 and got off at mile marker 57 at 3:15. What was my average speed on the turnpike? For a function f , the average rate of change of f over the interval [a, b] is given by f (b) − f (a) b − a If f (t) is my turnpike mile marker t hours after getting on, then the Pittsburgh example is the average rate of change of f over the interval [0, 2.25].

Example

A baseball thrown upward has height h(t) = 120t − 16t2 feet exactly t seconds after being thrown? What is the average speed

• f the ball over the first three seconds it is in the air? That is,

what is the average rate of change of h(t) over the interval [0, 3]?

Example

A baseball thrown upward has height h(t) = 120t − 16t2 feet exactly t seconds after being thrown? What is the average speed

• f the ball over the first three seconds it is in the air? That is,

what is the average rate of change of h(t) over the interval [0, 3]? The calculation for average rate of change is the same as computing the slope of the line connecting the points (0, h(0)) and (3, h(3)).

Example

Example

Slope = 216 − 0 3 − 0 = h(3) − h(0) 3 − 0