Transcript CH1 Section 1.3

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 Function
A function is a relationship between two variables such that
for each input there exists a unique output.
one and only one
input
rule
output
x=3
y = 5x
y = 15
Suppose input x = letter and output y = mailbox
This is a function:
x1
x2
x3
x4
y1
y2
y3
Each input has one and only one output.
This is not a function:
x1
y1
x2
y2
x3
y3
y4
There is an input with more than one output.
(Cannot deliver letter x3 to two different mailboxes!)
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.
p
-5
-3
6
7
12
q
-1
0
7
-2
0
True: For each input, there exists only one output
Different inputs can share the same output, like (–3, 0) and
(12, 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 can
repeat.)
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:
Year, x
Number of Visitors to a
Museum (in thousands), y
2004
2007
2010
2011
2013
85
68
124
178
205
Domain: {2004, 2007, 2010, 2011, 2013}
Range: {85, 68, 124, 178, 205}
7
Find the domain and range of the function: y 
x6
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 interval
notation (–, 0) U (0, ).
Find the domain and range of the function: y  3x  12
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 nonnegative
number, thus the outputs of this function are 0 or
positive.
y0
or
[0, )
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 make
an expression under a square root (or any even root)
negative.
Function Notation
It is always useful to give a function a name; the most
common 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 the
input 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 $350
deposit. The table below illustrates the total cost (in dollars)
of renting a dance room for different number of hours.
Hours, x
3
Total cost, f(x)
545
3.5
4.5
577.50 642.50
5
5.25
675
691.25
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) = 642.50
If the dance room is used 4.5 hours, the total cost is $642.50.
Using your textbook, practice the
problems assigned by your instructor to
review the concepts from Section 1.3.