Function Notation

• Evaluating a function means figuring out the value of a function’s output from a particular value of the input.

• Example. Let the function g be defined by:

Evaluate g(3) = ((3)2+1)/(5+3) = 10/8 = 1.25

.x5

1x g(x)

2

.a3

54a–a

2)–a(5

12)–(a 2)–g(a Evaluate

22

Evaluating functions using a table

• Suppose that f is defined by the table:

• To find f(3), we look in the table and get f(3) = –4.

• Now define g(x) = f(x+1). We evaluate g(3) = f(4) = –3. The table for g appears below:

• Why is no value listed for g(4)? Why is g defined at x= –2 while f is not?

x –1 0 1 2 3 4

f(x) 2 1 –5 2 –4 – 3

x –2 –1 0 1 2 3 4

g(x) 2 1 –5 2 –4 – 3 --

• Given an input, we evaluate a function to find the output. Often the situation is reversed; we know the output value and we want to find the corresponding input value(s). If the function is given by a formula, the input values are solutions to an equation.

• Problem. Let A = f(r) be the area of a circle of radius r, where r is in cm. What is the radius of a circle whose area is 100 cm2 ? Solution. The output f(r) is an area. Solving the equation f(r) = 100 for r gives us the radius of a circle whose area is 100 cm2. Since the formula for the area of a circle is

we solve This yields and we take the positive

value.

,r f(r) 2 .100r f(r) 2 5.64, r

Finding input and output values using a table

• Suppose that f is defined by the table:

• As before, to find f(1), we look in the table and get f(1) = –5.

• Now suppose we want to solve f(x) = 2 for x.

There are two values for x which satisfy this condition,

namely, x = –1 and x = 2.

x –1 0 1 2 3 4

f(x) 2 1 –5 2 –4 – 3

Changes in input and output

• For the function Q = f(t), a change inside the function’s parentheses can be called an “inside change” and a change outside the function’s parentheses can be called an “outside change”.

• Problem. For the function, which represents the area of a circle of radius r, contrast the expressions f(r + 10) and f(r) + 10. Solution. f(r + 10) is the area of a circle of radius = (r + 10). f(r) + 10 is the area of a circle of radius = r plus 10 more units of area.

,r f(r) 2

Input and output from a graph showing an influenza epidemic

• We have I = f(w), where I is the number of individuals infected (in thousands) w weeks after the epidemic begins.

• Evaluate f(2) and explain its meaning.• Solve f(w) = 4.5 and explain the meaning of the solution.

Domain and Range

• Assume that Q = f(t). The domain of f is the set of input values, t, which yield an output value. The range of f is the corresponding set of output values.

• There are two types of reasons that the domain of a function f may be restricted: (1)The real-world situation being modeled by f does not make sense without a restriction. (2)The formula used to define f does not make sense without a restriction (often because of division by 0).

• If the domain of a function is not specified, we usually assume that it is as large as possible--that is, all numbers that make sense as inputs for the function.

• Problem. Find the domain and range of

Solution. Since no restrictions on the domain are given, we make it as large as possible. That is, all real numbers except 0.

To determine the range, we suppose that y is in the range so that

The latter equation can be solved for x if Also, if y = 2, the latter equation has no solution for x. Thus, the range of f is the set of all real numbers except 2.

.x

1 2 f(x) y

or ,x

1 2 y

.x

1 2–y

2.y

Use of Maple to plot function of previous slide

> plot(2+(1/x),x=-3..3,-10..10,color=black);

Finding the domain and range for a function whose graph is given

Domain

Range

Model for height of a sunflower plant

• The height of a certain sunflower in centimeters is:

• If the domain of h is , where t is in days, the graph of h is:

• By examining the graph, can you tell what the range of h is?

t24(0.9)1

260h(t)

t0

Piecewise Defined Functions• A function may employ different formulas on different parts

of its domain. Such a function is said to be piecewise defined. Note that we say there is one function even though several formulas may be used.

• Example. A long-distance calling plan charges 99 cents for any call up to 20 minutes in length and 7 cents a minute for each additional minute with part of a minute pro-rated. Let C be the cost of a call in cents as a function of its length t in minutes. We have C = f(t), where the formulas for this function are:

The graph of this function is shown on the next slide.

20. for t 20)7(t 99

20 t 0for 99 f(t) C

0 for t0

Cost Function for Long Distance Calling Plan

>plot(piecewise(t<=0,0,t<=20,99,t>20,7*t–41),t=–10..35,C=0..200,color=black);

• Where is the point on the graph when t = 0?

• The next problem introduces a piecewise defined function which is used to round a value x down to the nearest integer. How would you define the function that rounds a value x up to the nearest integer, and what would you call it?

• Problem. For any real number x, we define the floor of x, denoted as the greatest integer that does not exceed x. Find the formulas for the floor of x when

Solution.

Note that the floor function is first described in words, and this yields an infinite number of different formulas for the function.

103. x 100

100. x have we101, x 100

,x

.102x have we103,x102For .101x have we,102x101For

For

Use of Maple to graph the floor function

>plot(floor(x),x=100..103,color=black,discont=true);

The Absolute Value Function• The absolute value function is a piecewise defined

function defined by

• The graph of y = | x | is shown below:

• The absolute value function is used to measure the distance |x–y| between points x and y on the number line. For example, the distance from –1 to 2 is | –1 –2| = | –3| = 3.

0. x x,

0 x x, |x| f(x)

Suppose an employee is paid $5.00 per hour to work a standard 40 hour work week. If he works overtime, he is paid $7.50 per hour up to a maximum of 80 hours. The graph below shows his weekly pay as a function, f(t), of the time worked.

p2

p1

What are the values of p1 and p2?

Pay function in bracket form(80,500)

(40,200)

f(t) = 80t40 100,7.5t

40t,0 t 5

Composite Functions

• Two functions may be combined by requiring the output of one to be the input of the other.

• For example, let n = f(A) =A/250 be the number of gallons of paint needed to paint A square feet, and let C = g(n) = 30.5n be the cost, in dollars, of n gallons of paint.

• Substituting n into the formula for C, we obtain where the function h is said to be the composition of g with f.

• We say that f is the inside function and g is the outside function.

A,122.0250

A30.5g(f(A))h(A)C

Example of composite functions• Let f(x) = |x| – 1. The graph of f is shown next.

• Let g(x) = |x|. Then h(x) = g(f(x)) = | |x| – 1 |. Then the graph of h(x) is:

• Challenge: Draw the graph of h(h(x)).

Interchanging input and output: Inverse Functions• The roles of input and output are not necessarily fixed. In

an earlier example, we derived a function f which converts degrees Celsius, C, to degrees Fahrenheit, F. The formula for f was:

Suppose now that we know the value of F and we wish to compute the value of C. We can define a new function g such that C = g(F). For this function, F is the input and C is the output. The functions f and g are called inverses of each other.

• Find a formula for the inverse function C = g(F). We solve the previous equation for C to obtain:

.32 C5

9f(C) F

.)32– (F9

5g(F) C

Inverse Function Notation

• In the preceding discussion on temperature conversion, their was nothing about the names of the two functions that stressed their special relationship. If we want to emphasize that a function is the inverse of f, we call it f -1, which is read “f-inverse”.

• For temperature conversion, we have:

• Warning:

.)32– (F9

5(F)f C 1-

different! are f

1 and f 1-

A function and its inverse "undo" each other.

• What happens if we convert from Celsius to Fahrenheit and then back to Celsius? Answer: We are back where we started. In terms of function composition,

• If, on the other hand, we convert from Fahrenheit to Celsius and then back to Fahrenheit, we have

C(f(C))f -1

F(F))f(f -1

A function and its inverse "undo" each other, continued.

• What happens if we convert from Celsius to Fahrenheit and then back to Celsius?

• If, on the other hand, we convert from Fahrenheit to Celsius and then back to Fahrenheit, we have

C)3232C5

9(

9

532)f(C)(

9

5(f(C))f 1-

F32))32F(9

5(

5

932(F)f

5

9(F))f(f 1-1-

Finding a formula for an inverse function

• In the graph below, we have P = f(t) = 20 + 0.4t, and we want to graphically find

P

(25).f -1

Locate 25 on the P-axis

Read off the value of t corresponding to P = 25.

• If we continue with the example from the previous slide, we may also solve algebraically for The equation to be solved is: 20 + 0.4t = 25, and we first subtract 20 from both sides to obtain 0.4t = 5. Upon dividing by 0.4, we have t = 12.5 .

• This same algebraic procedure can be carried for a general P to obtain the formula for We must solve 20 + 0.4t = P for t. We have 0.4t = P – 20, and thus,

That is,

(25).f -1

.f -1

50. P5.20.4

20P t

50. P5.2P)(f -1

Facts about Inverse Functions.

• Suppose f has an inverse function. Then outputs of f are inputs of f -1. Similarly, outputs from f -1 are inputs of f. It follows that:

Domain of f -1 = Range of f and

Range of f -1 = Domain of f.

• Warning: Not all functions have inverses. The functions which have inverses are called invertible.

• Example. The absolute value function y = | x | is not invertible. Do you see why?

Concavity

• Suppose we graph Q = f(t). If the graph bends upward as t increases, we say that f or its graph is concave up. If the graph bends downward as t increases, we say that f or its graph is concave down.

• The average rate of change of f over small intervals is what determines the concavity of f.

• If the average rate of change of f increases, then f is concave up.

• If the average rate of change of f decreases, then f is concave down.

• Recall Karim’s bike trip. What can you say about concavity for Karim? Was his average speed increasing or decreasing? What about Amanda’s bike trip?

Increasing and Decreasing Functions; Concavity• Increasing, Concave Down Decreasing, Concave Down

• Decreasing, Concave Up Increasing, Concave Up

Summary for Functions

• Evaluating a function means figuring out the value of a function’s output from a particular value of the input.

• Sometimes we are given the output value and we must solve for the input value.

• Important terms: “inside change” and “outside change”.

• Assume that Q = f(t). • The domain of f is the set of inputs which yield an output value. • The range of f is the corresponding set of output values.

• A function which uses different formulas in different parts of its domain is said to be piecewise defined.

• If h(x) = f(g(x)), we say that h is the composition of f with g.

• If we are given y = f(x), and the roles of inputs and outputs are reversed, we have the inverse function x = f -1(y). Warning: Not all functions have inverses.

Summary for Functions, continued

• When a function is composed with its inverse, we have

• We discussed concavity of the graph of a function f and its relation to the rate of change of f.

x.x))(f(f and x (f(x))f -1-1