1. 1-7 Function Notation Aim: To write functions using function notation.
To evaluate and graph functions.

2. Rules
Some relationships between ordered pairs can be described using rules.
For example, the equation y = 3x is a rule stating that the output is three times the input.
When writing equations, x is the input, or independent variable, and y is the output, or dependent variable.

3. Example 1
Study the relationship between x and y. Then write an equation that relates the two. x y 1 2 5 10 7 14
y = 2x

4. Example 2
Study the relationship between x and y. Then write an equation that relates the two. x y –4 –1 2 3 5 8
y = x + 3

5. Example 3
Study the relationship between x and y. Then write an equation that relates the two. x y –3 –5 –1 2 5 3 7
y = 2x + 1

6. Function Rules
When a rule describes the relationship between the points of a function, the equation can be written using function notation.
An algebraic expression that defines a function is a function rule.
A function rule defines the output based upon the input.

7. Example 4
A loaf of bread costs $2. Use x and y to write a function rule for the cost of any number of loaves of bread.
Remember: x is the independent variable and y is the dependent variable.
x = number of loaves of bread purchased
y = cost of x loaves of bread
Cost =
Price per loaf •
Number of loaves
y = 2x

8. Function Notation
In function notation, the dependent variable, y, is replaced with a term such as f (x).
The letter f is the name of the function and (x) indicates that the output is dependent upon the input, x.
In other words, the output is defined in terms of the input, x.
f (x) is the output and is therefore interchangeable with y.

9. Function Notation f (x) is read “f of x.”
f (2) is read “f of 2” and means the output of function f for an input of 2.
Any letter can be used to name a function, although f, g, and h are most common.
g(4) is read “g of 4” and means the output of function g for an input of 4.
f (x) ≠ “f times x”

10. Variables Of A Function
There are several different ways to describe the variables of a function.
Independent Variable
Dependent Variable
x-values
y-values
Domain
Range
Input
Output x
f (x)

11. f (x) = 2x f (b) = 2b Example 4b
A loaf of bread costs $2. Use function notation to represent the cost of any number of loaves of bread, x.
Use function notation to represent the cost of any number of loaves of bread, b.
f (x) = 2x
f (b) = 2b

12. Example 5
Weekend admission to the Big E is $15. Each cream puff that you eat costs $3. Write a function that determines the total cost of attending the Big E.
x = number of cream puffs purchased
Cost =
Admission +
Cost per c.p. •
# of c.p.
f (x) = x

13. Evaluating Functions
To evaluate a function means to plug in an input and find the corresponding output, which is usually a number.

14. f (x) = x – 5 g(x) = 2x h(x) = –x + 2 f (8) = 3 g(8) = 12 h(8) = –6
Example 6
Evaluate the functions for the given values.
f (x) = x – 5
g(x) = 2x
h(x) = –x + 2
f (8) = 3
f( b-3)
g(8) = 12
h(8) = –6

15. f (x) = 2.5x – 2 g(x) = 9 h(x) = f (1) = 0.5 f (4) = 8 g(3) = 9
Example 7
Evaluate the functions for the given values.
f (x) = 2.5x – 2
g(x) = 9
h(x) =
f (1) =
0.5
f (4) = 8
g(3) = 9
g(1.7) = 9
h(10) = 4
h(–5) = –1

16. Graphs Of Functions
The graph of a function is a picture of the function’s ordered pairs, or points.
To graph a function, you can make a table of values for the function, plots these points, and connect the points to complete the pattern.
Sometimes it might be stated or implied that the function only exists for a certain domain, in which case only plot and connect the points within this domain.

17. Example 8
The graph of function f is shown. Use the graph to evaluate the function for the given values. 1 2 3 4 5 6 7 8
f (2) = 4
f (3) = 5
f (7) = 4

18. Example 9
The graph of function f is shown. Use the graph to evaluate the function for the given values. 1 2 3 4 5 6 7 8
f (1) = 2
f (5) = 5
f (7) = 1

19. Example 10
Graph the function.
f (x) = 2x – 1 x y –1 1 2 –3 –1 1 3

20. f (x) = –2x – 2 for –3 < x < 1
Example 11
Graph the function.
f (x) = –2x – 2 for –3 < x < 1 x y –2 –1 1 2 –2 –4