Holt CA Course Functions Preparation for AF3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of a graph. California Standards

Holt CA Course Functions A function is a set of ordered pairs (x, y) so that each x-value corresponds to exactly one y-value. Some function can be described by a rule written in words, such as “double a number and then add nine to the result,” or by an equation with two variables. One variable (often x) represents the input, and the other variable (often y) represents the output. Output variable Input variable Function Rule

Holt CA Course Functions When a function can be written as an equation, the input is the value substituted into the function rule. The output is the result of that substitution.

Holt CA Course Functions Additional Example 1: Finding Output Values Substitute –2 for x. Then simplify. Substitute 0 for x. Then simplify. Substitute 2 for x. Then simplify. Find the output for each input: –2, 0, 2. Input A. y = 3 – x 2 Rule Output x 3 – x 2 y – – (–2) 2 3 – (0) 2 3 – (2) 2 –1 3 Make a table.

Holt CA Course Functions The input values of a function are also called the domain. The output values of a function are also called the range. Reading Math

Holt CA Course Functions Additional Example 1: Completing a Function Table Substitute –2 for x and simplify. Substitute 0 for x and simplify. Substitute 2 for x and simplify. Find the output for each input: – 2, 0, 2. Input B. y = –2x - 3 Rule Output x –2x – 3 y – –3 –7 Make a table. –2(–2) – 3 –2(0) – 3 –2(2) – 3

Holt CA Course Functions Check It Out! Example 1 Substitute –2 for x. Then simplify. Substitute 0 for x. Then simplify. Substitute 5 for x. Then simplify. Find the output for each input: –2, 0, 5 Input A. y = 5x + 3 Rule Output x 5x + 3 y – (–2) + 3 5(0) + 3 5(5) + 3 – Make a table.

Holt CA Course Functions Check It Out! Example 1 Substitute –2 for x and simplify. Substitute 0 for x and simplify. Substitute 5 for x and simplify. Find the output for each input: –2, 0, 5. Input B. y = 3x 2 Rule Output x 3x23x2 y – (–2) 2 3(0) 2 3(5) Make a table.

Holt CA Course Functions Because a function has exactly one output for each input, you can use the vertical line test to determine whether a graph is a function. If no vertical line intersects the graph at more than one point, then the graph is a function. One way to perform the vertical line test is to pass a vertical line across a graph.

Holt CA Course Functions Determine if the relationship represents a function. A. Additional Example 2: Identifying Functions The input x = 2 has two outputs, y = 3 and y = 6. The input x = 3 also has more than one output. xy The relationship is not a function.

Holt CA Course Functions Determine if the relationship represents a function. B. Additional Example 2: Identifying Functions Each input has only one output value. xy –1 2–4 5– 7 8–10 The relationship is a function.

Holt CA Course Functions Determine if the relationship represents a function. C. Additional Example 2: Identifying Functions Pass a vertical line across the graph. Many vertical lines intersect the graph at two points. The relationship is not a function.

Holt CA Course Functions Determine if the relationship represents a function. D. The relationship is a function. Additional Example 2: Identifying Functions x y Pass a vertical line across the graph. No vertical lines intersect the graph at more than one point.

Holt CA Course Functions Determine if the relationship represents a function. A. Check It Out! Example 2 Each input x has only one output y. xy The relationship is a function.

Holt CA Course Functions Determine if the relationship represents a function. B. Each input has only one output value. xy The relationship is a function. Check It Out! Example 2

Holt CA Course Functions Determine if the relationship represents a function. C. Check It Out! Example 2 Pass a vertical line across the graph. No vertical lines intersect the graph at more than one point. x y The relationship is a function

Holt CA Course Functions Determine if the relationship represents a function. D. The relationship is a function. Pass a vertical line across the graph. No vertical lines intersect the graph at more than one point. Check It Out! Example 2 x y