9 5 Functions and Their Inverses Warm Up

9 -5 Functions and Their Inverses Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

9 -5 Functions and Their Inverses Warm Up Solve for x in terms of y. 1. 2. 3. 4. y = 2 ln x Holt Algebra 2

9 -5 Functions and Their Inverses 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. Holt Algebra 2

9 -5 Functions and Their Inverses Vocabulary inverse function inverse relation one-to-one function Holt Algebra 2

9 -5 Functions and Their Inverses You have seen the word inverse used in various ways. The additive inverse of 3 is – 3. The multiplicative inverse of 5 is Holt Algebra 2

9 -5 Functions and Their Inverses You can also find 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 yvalues in each ordered pair of the relation. Remember! 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. Holt Algebra 2

9 -5 Functions and Their Inverses Example 1: Graphing Inverse Relations Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation. Graph each ordered pair and connect them. Switch the x- and y-values in each ordered pair. x y Holt Algebra 2 2 0 5 1 6 5 9 8 x 0 1 5 8 y 2 5 6 9 • • • ● •

9 -5 Functions and Their Inverses Check It Out! Example 1 Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation. Graph each ordered pair and connect them. x 1 3 4 5 6 y 0 1 2 3 5 Switch the x- and y-values in each ordered pair. x y Holt Algebra 2 0 1 1 3 2 4 3 5 5 6 • • •

9 -5 Functions and Their Inverses 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). Holt Algebra 2

9 -5 Functions and Their Inverses 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. Holt Algebra 2

9 -5 Functions and Their Inverses Holt Algebra 2

9 -5 Functions and Their Inverses Example 1 A: 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. Holt Algebra 2

9 -5 Functions and Their Inverses Example 1 B: 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. Holt Algebra 2

9 -5 Functions and Their Inverses 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. Holt Algebra 2

9 -5 Functions and Their Inverses 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. Holt Algebra 2

9 -5 Functions and Their Inverses 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. Holt Algebra 2

9 -5 Functions and Their Inverses 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. Holt Algebra 2

9 -5 Functions and Their Inverses 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 Holt Algebra 2 R}.

9 -5 Functions and Their Inverses Check It Out! Example 2 Find the inverse of f(x) = 2 x – 4. Determine whether it is a function, and state its domain and range. Holt Algebra 2

9 -5 Functions and Their Inverses Check It Out! Example 2 Continued Step 1 Find the inverse. y = 2 x – 4 Rewrite the function using y instead of f(x). x = 2 y – 4 Switch x and y in the equation. x + 4= 2 y Add 2 to both sides of the equation. 2 y = x+4 Reverse the sides. Divide by 2 and Simplify. Holt Algebra 2

9 -5 Functions and Their Inverses 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. Holt Algebra 2

9 -5 Functions and Their Inverses 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. Holt Algebra 2

9 -5 Functions and Their Inverses Holt Algebra 2

9 -5 Functions and Their Inverses Example 3: Determining Whether Functions Are Inverses Determine by composition whether each pair of functions are inverses. f(x) = 3 x – 1 and g(x) = 1 3 x+1 Find the composition f(g(x)) = 3( Holt Algebra 2 1 3 x + 1) – 1 Substitute x in f. 1 3 x + 1 for = (x + 3) – 1 Use the Distributive Property. =x+2 Simplify.

9 -5 Functions and Their Inverses 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. Holt Algebra 2

9 -5 Functions and Their Inverses Example 3 B: Determining Whether Functions Are Inverses For x ≠ 1 or 0, f(x) = 1 x– 1 and g(x) = 1 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. Holt Algebra 2

9 -5 Functions and Their Inverses Example 3 B Continued Check The graphs are symmetric about the line y = x for all x but 0 and 1. Holt Algebra 2

9 -5 Functions and Their Inverses Check It Out! Example 3 a Determine by composition whether each pair of functions are inverses. f(x) = 2 3 x + 6 and g(x) = 3 2 x– 9 Find the composition f(g(x)) and g(f(x)). f(g(x)) = 2 3 ( 3 2 x – 9) + 6 g(f(x)) = 3 2 ( 2 3 x + 6) – 9 =x– 6 +6 =x+9 – 9 =x =x Because f(g(x)) = g(f(x)) = x, they are inverses. Holt Algebra 2

9 -5 Functions and Their Inverses Check It Out! Example 3 a Continued Check The graphs are symmetric about the line y = x for all x. Holt Algebra 2

9 -5 Functions and Their Inverses Check It Out! Example 3 b f(x) = x 2 + 5 and for x ≥ 0 Find the compositions f(g(x)) and g(f(x)). f(g(x)) = +5 = x -10 x + 25 +5 = x – 10 x + 30 Holt Algebra 2 Substitute in f. Simplify. for x

9 -5 Functions and Their Inverses Check It Out! Example 3 b 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. Holt Algebra 2

9 -5 Functions and Their Inverses 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 Holt Algebra 2

9 -5 Functions and Their Inverses Lesson Quiz: Part II 2. Find the inverse f(x) = x 2 – 4. Determine whether it is a function, and state its domain and range. not a function D: {x|x ≥ 4}; R: {all Real Numbers} Holt Algebra 2

9 -5 Functions and Their Inverses Lesson Quiz: Part III 3. Determine by composition whether f(x) = 3(x – 1)2 and g(x) = for x ≥ 0. yes Holt Algebra 2 +1 are inverses