6 6 Functions and Their Inverses Warm Up

6 -6 Functions and Their Inverses Warm Up Lesson Presentation Lesson Quiz Holt. Mc. Dougal Algebra 2 Holt

6 -6 Functions and Their Inverses Warm Up Solve for x in terms of y. 1. 2. 3. 4. y = 2 ln x Holt Mc. Dougal Algebra 2

6 -6 Functions and Their Inverses Objectives Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Holt Mc. Dougal Algebra 2

6 -6 Functions and Their Inverses Vocabulary one-to-one function Holt Mc. Dougal Algebra 2

6 -6 Functions and Their Inverses In Lesson 4 -2, you learned that the inverse of a function f(x) “undoes” f(x). Its graph is a reflection across line y = x. The inverse may or not be a function. 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 Mc. Dougal Algebra 2

6 -6 Functions and Their Inverses Holt Mc. Dougal Algebra 2

6 -6 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 Mc. Dougal Algebra 2

6 -6 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 Mc. Dougal Algebra 2

6 -6 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 Mc. Dougal Algebra 2

6 -6 Functions and Their Inverses Recall from Lesson 4 -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 Mc. Dougal Algebra 2

6 -6 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 The horizontal-line test shows that the inverse is a function. Note that the domain and range of f are all real numbers. Holt Mc. Dougal Algebra 2

6 -6 Functions and Their Inverses Example 2 Continued Step 1 Find the inverse. Rewrite the function using y instead of f(x). Switch x and y in the equation. Cube both sides. Simplify. Isolate y. Holt Mc. Dougal Algebra 2

6 -6 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 R}. Check Graph both relations to see that they are symmetric about y = x. Holt Mc. Dougal Algebra 2

6 -6 Functions and Their Inverses Check It Out! Example 2 Find the inverse of f(x) = x 3 – 2. Determine whether it is a function, and state its domain and range. 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 Mc. Dougal Algebra 2

6 -6 Functions and Their Inverses Check It Out! Example 2 Continued Step 1 Find the inverse. y = x 3 – 2 Rewrite the function using y instead of f(x). x = y 3 – 2 Switch x and y in the equation. x + 2 = y 3 Add 2 to both sides of the equation. 3 x + 2 = 3 y 3 3 x+2=y Holt Mc. Dougal Algebra 2 Take the cube root of both sides. Simplify.

6 -6 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. Check Graph both relations to see that they are symmetric about y = x. Holt Mc. Dougal Algebra 2

6 -6 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 Mc. Dougal Algebra 2

6 -6 Functions and Their Inverses Holt Mc. Dougal Algebra 2

6 -6 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( 1 3 x + 1) – 1 Substitute x in f. 1 3 x + 1 for = (x + 3) – 1 Use the Distributive Property. =x+2 Simplify. Holt Mc. Dougal Algebra 2

6 -6 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 Mc. Dougal Algebra 2

6 -6 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 Mc. Dougal Algebra 2

6 -6 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 Mc. Dougal Algebra 2

6 -6 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 Mc. Dougal Algebra 2

6 -6 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 Mc. Dougal Algebra 2

6 -6 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 Mc. Dougal Algebra 2 Substitute in f. Simplify. for x

6 -6 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 Mc. Dougal Algebra 2

6 -6 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 Mc. Dougal Algebra 2

6 -6 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 Mc. Dougal Algebra 2

6 -6 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 Mc. Dougal Algebra 2 +1 are inverses