Chapter 1 Linear Functions and Mathematical Modeling

Section 1.3

Section 1.3 Functions: Definition, Notation, and

Evaluation

• Definition of Function

• Verbal, Numeric, Symbolic, and Graphical Descriptions of Functions

• Domain and Range

• Function Notation

• Evaluating Functions in Different Contexts

Definition of a FunctionA function is a relationship between two variables such that for each input there exists a unique output.

one and only one

ruleoutput

y = 5x

y = 15

input

x = 3

Suppose input x = letter and output y = mailbox

This is a function:

Each input has one and only one output.

This is not a function:

There is an input with more than one output. (Cannot deliver letter x3 to two different mailboxes!)

4

3

2

1

x

x

x

x

3

2

1

x

x

x

3

2

1

y

y

y

4

3

2

1

y

y

y

y

Example of a Function Verbal Description: The total salary for a math tutor will be calculated by multiplying the number of hours worked times the hourly rate.

Numeric (Tabular) Description:

Suppose the tutor earns $15 per hour.

Number of hours worked, x 10 15 22 35 40

Salary in dollars, y 150 225 330 525 600

Example of a Function (contd.) Symbolic Description: y = 15x where x represents number of hours worked, and y is the tutor’s total salary.

Graphical Description:

True or False: The following table represents a function.

True: For each input, there exists only one output

Different inputs can share the same output, like (–3, 0) and (12, 0)

p -5 -3 6 7 12

q -1 0 7 -2 0

Are these functions?

x -1 0 1 2

y 6 4 -1 -7

x -1 0 1 2

y 5 0 5 12

x 0 4 9 4

y -8 -6 -3 2

Yes; each input (or x-value) has one and only one output (or y-value).

Yes; each input has one and only one output. (Reminder: x-values cannot repeat, but y-values canrepeat.)

No; input “4” is repeated. x = 4 does not have a unique (one and only one) y-value.

Domain and Range of a Function Domain: Possible values of the input. Range: Possible values of the output.

Find the domain and range of the following function:

Domain: {2004, 2007, 2010, 2011, 2013} Range: {85, 68, 124, 178, 205}

 

Year, x 2004 2007 2010 2011 2013

Number of Visitors to a Museum (in thousands), y

85 68 124 178 205

Find the domain and range of the function:

Domain:Since division by zero is undefined (it is mathematically impossible), the function is undefined for all the x-values that yield a 0 in the divisor (denominator). So, we must exclude any x-values that result in division by zero.

Therefore, the domain is the set of all real numbers, with one restriction, x ≠ 6. In interval notation, (–, 6) U (6,).

Range:The only way the given function would equal zero is to have a numerator 0, which is not the case.

The range is the set of all real numbers except 0. In intervalnotation (–, 0) U (0, ).

 

6x7

y

Find the domain and range of the function: Domain: The square root of a number is a real number only if the radicand is nonnegative.

3x – 12 0

3x 12

x 4 or [4, )

Range: Taking the square root will result in a nonnegativenumber, thus the outputs of this function are 0 or positive.

y 0 or [0, )

12x3y

Main Restrictions for Domain

1. Exclude any x-values that result in division by zero. 

2. Exclude any x-values that result in even roots of negative numbers. That is, any x-values which makean expression under a square root (or any even root) negative.

 

Function Notation

It is always useful to give a function a name; the mostcommon name is “f.” We can also use other symbols or letters like g, h, p, etc.

If x represents an input and y represents the corresponding output, the function notation is given by f(x) = y.

That is:f(x) is the output for the function f when the input is x.

f(input) = output.

f(x) is read “f of x” or “the value of f at x.”

Note: In function notation, f(x) does not mean multiplication of f times x.

 

Evaluating a Function

If f(x) = 2x² – x + 3, find f(–5)

f(–5) means to find the value of the function when theinput variable has a value of –5.

Substitute –5 for x and simplify.

f(–5) = 2(–5)² – (–5) + 3

= 2(25) + 5 + 3

= 50 + 5 + 3

= 58

f(–5) = 58; that is, when the input is –5, the output is 58.

If f(x) = x² – 3x + 4, find f(a).

Substitute “a” for x and simplify.

f(a) = (a)² – 3(a) + 4

= a² – 3a + 4

Since we have no numerical value for a, we stop!

If f(x) = x² – 3x, find f(a + 2).

Substitute “a + 2” for x and simplify.

f(a + 2) = (a + 2)² – 3(a + 2)

= (a + 2)(a + 2) – 3(a + 2)

= a² + 4a + 4 – 3a – 6

= a² + a – 2

Caution: (a + 2)² ≠ a² + 4

Dance Rooms Here charges $65 per hour and a $350deposit. The table below illustrates the total cost (in dollars) of renting a dance room for different number of hours.

a. Find the value(s) of x when f(x) = 675.We want to find the input when the output is 675. Therefore, x = 5.

b. Find and interpret f(4.5). We want to find the output when the input is 4.5. Therefore, f(x) = 642.50If the dance room is used 4.5 hours, the total cost is $642.50.

Hours, x 3 3.5 4.5 5 5.25

Total cost, f(x) 545 577.50 642.50 675 691.25

Using your textbook, practice the problems assigned by your instructor to review the concepts from Section 1.3.