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

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.

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

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

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

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.

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

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.

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”

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)

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

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) = 15 + 3x

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

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

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

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.

Example 8 The graph of function f is shown. Use the graph to evaluate the function for the given values. 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 f (2) = 4 f (3) = 5 f (7) = 4

Example 9 The graph of function f is shown. Use the graph to evaluate the function for the given values. 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 f (1) = 2 f (5) = 5 f (7) = 1

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

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