Functions and Their Inverses 9-5 Functions and Their Inverses Warm Up Lesson Presentation Lesson Quiz Holt Algebra2

Warm Up Solve for x in terms of y. 1. 2. 3. 4. y = 2ln x

Objectives Graph and recognize inverses of relations and functions. Find inverses of functions. Determine whether the inverse of a function is a function. Write rules for the inverses of functions.

Vocabulary inverse function inverse relation one-to-one function

You have seen the word inverse used in various ways. The additive inverse of 3 is –3. The multiplicative inverse of 5 is

You can also find and apply inverses to relations and functions You can also find and apply inverses to relations and functions. To graph the inverse relation, you can reflect each point across the line y = x. This is equivalent to switching the x- and y-values in each ordered pair of the relation. A relation is a set of ordered pairs. A function is a relation in which each x-value has, at most, one y-value paired with it. Remember!

Example 1: Graphing Inverse Relations Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation. x 1 5 8 y 2 6 9 Graph each ordered pair and connect them. ● • • Switch the x- and y-values in each ordered pair. ● ● • • • x 2 5 6 9 y 1 8 ● • • •

1 3 4 5 6 2 • • • • • • • • • • • • • • • Check It Out! Example 1 Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation. x 1 3 4 5 6 y 2 Graph each ordered pair and connect them. Switch the x- and y-values in each ordered pair. • • • • • • • • x 1 2 3 5 y 4 6 • • • • • • •

When the relation is also a function, you can write the inverse of the function f(x) as f–1(x). This notation does not indicate a reciprocal. Functions that undo each other are inverse functions. To find the inverse function, use the inverse operation. In the example above, 6 is added to x in f(x), so 6 is subtracted to find f–1(x).

Recall that the vertical-line test (Lesson 1-6) can help you determine whether a relation is a function. Similarly, the horizontal-line test can help you determine whether the inverse of a function is a function.

Example 1A: Using the Horizontal-Line Test Use the horizontal-line test to determine whether the inverse of the blue relation is a function. The inverse is a function because no horizontal line passes through two points on the graph.

Example 1B: Using the Horizontal-Line Test Use the horizontal-line test to determine whether the inverse of the red relation is a function. The inverse is a not a function because a horizontal line passes through more than one point on the graph.

Check It Out! Example 1 Use the horizontal-line test to determine whether the inverse of each relation is a function. The inverse is a function because no horizontal line passes through two points on the graph.

Recall from Lesson 7-2 that to write the rule for the inverse of a function, you can exchange x and y and solve the equation for y. Because the value of x and y are switched, the domain of the function will be the range of its inverse and vice versa.

Example 2: Writing Rules for inverses Step 1 The horizontal-line test shows that the inverse is a function. Note that the domain and range of f are all real numbers.

Example 2: Writing Rules for inverses Find the inverse of . Determine whether it is a function, and state its domain and range. Step 1 Find the inverse. Rewrite the function using y instead of f(x). Switch x and y in the equation. Solve for y. Simplify. Isolate y.

Example 2 Continued Because the inverse is a function, . The domain of the inverse is the range of f(x):{x|x R}. The range is the domain of f(x):{y|y R}.

Check It Out! Example 2 Find the inverse of f(x) = 2x – 4. Determine whether it is a function, and state its domain and range.

Check It Out! Example 2 Continued Step 1 Find the inverse. y = 2x – 4 Rewrite the function using y instead of f(x). x = 2y – 4 Switch x and y in the equation. x + 4= 2y Add 2 to both sides of the equation. 2y = x+4 Reverse the sides. Divide by 2 and Simplify.

Check It Out! Example 2 Continued Because the inverse is a function, . The domain of the inverse is the range of f(x): R. The range is the domain of f(x): R.

You have seen that the inverses of functions are not necessarily functions. When both a relation and its inverses are functions, the relation is called a one-to-one function. In a one-to-one function, each y-value is paired with exactly one x-value. You can use composition of functions to verify that two functions are inverses. Because inverse functions “undo” each other, when you compose two inverses the result is the input value x.

Example 3: Determining Whether Functions Are Inverses Determine by composition whether each pair of functions are inverses. 1 3 f(x) = 3x – 1 and g(x) = x + 1 Find the composition f(g(x)). Substitute x + 1 for x in f. 1 3 f(g(x)) = 3( x + 1) – 1 1 3 Use the Distributive Property. = (x + 3) – 1 = x + 2 Simplify.

Example 3 Continued Because f(g(x)) ≠ x, f and g are not inverses. There is no need to check g(f(x)). Check The graphs are not symmetric about the line y = x.

Example 3B: Determining Whether Functions Are Inverses For x ≠ 1 or 0, f(x) = and g(x) = + 1. 1 x x – 1 Find the compositions f(g(x)) and g(f (x)). = (x – 1) + 1 = x = x Because f(g(x)) = g(f (x)) = x for all x but 0 and 1, f and g are inverses.

Example 3B Continued Check The graphs are symmetric about the line y = x for all x but 0 and 1.

Determine by composition whether each pair of functions are inverses. Check It Out! Example 3a Determine by composition whether each pair of functions are inverses. 3 2 f(x) = x + 6 and g(x) = x – 9 Find the composition f(g(x)) and g(f(x)). f(g(x)) = ( x – 9) + 6 3 2 g(f(x)) = ( x + 6) – 9 2 3 = x – 6 + 6 = x + 9 – 9 = x = x Because f(g(x)) = g(f(x)) = x, they are inverses.

Check It Out! Example 3a Continued Check The graphs are symmetric about the line y = x for all x.

10 x - Check It Out! Example 3b f(x) = x2 + 5 and for x ≥ 0 Find the compositions f(g(x)) and g(f(x)). f(g(x)) = + 5 Substitute for x in f. = x + 25 +5 10 x - Simplify. = x – 10 x + 30

Check It Out! Example 3b Continued Because f(g(x)) ≠ x, f and g are not inverses. There is no need to check g(f(x)). Check The graphs are not symmetric about the line y = x.

Lesson Quiz: Part I 1. Use the horizontal-line test to determine whether the inverse of each relation is a function. A: yes; B: no

Lesson Quiz: Part II 2. Find the inverse f(x) = x2 – 4. Determine whether it is a function, and state its domain and range. not a function D: {x|x ≥ 4}; R: {all Real Numbers}

Lesson Quiz: Part III 3. Determine by composition whether f(x) = 3(x – 1)2 and g(x) = +1 are inverses for x ≥ 0. yes