Use Inverse Functions Lesson 3.4 Honors Algebra 2 Use Inverse Functions Lesson 3.4

Goals Goal Rubric Find an Inverse Relation. Verify that Functions are Inverses. Find the Inverse of a Power function. Find the Inverse of a Cubic Function. Find the Inverse of a Power Model. Use an Inverse Power Model to Make a Prediction. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary Inverse Relation Inverse Function

Inverse Relations You have seen the word inverse used in various ways. The additive inverse of 3 is –3. The multiplicative inverse of 5 is 1 5 Inverse operations, such as; addition and subtraction

Inverse Relations You can also find and apply inverses to relations and functions. Recall that a relation is a pairing of the input (x) values with output (y) values. An inverse relation interchanges the input (x) and output (y) values of the original relation. This means that the domain and range are also interchanged. 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!

Inverse Relations The graph of an inverse relation is a reflection of the graph of the original relation over the line y = x. • • • • • Original Relation • x 1 5 8 y 2 6 9 • • Inverse Relation x 2 5 6 9 y 1 8 Domain Original → Range Inverse Range Original → Domain Inverse

Inverse Relation To find the inverse of a relation algebraically, interchange x and y and solve for y. y = inverse relation Solve for y Trade x and y places

Find an inverse relation EXAMPLE 1 Find an inverse relation Find an equation for the inverse of the relation y = 3x – 5. y = 3x – 5 Write original relation. x = 3y – 5 Switch x and y. x + 5 = 3y Add 5 to each side. 1 3 x + 5 = y Solve for y. This is the inverse relation.

Inverse Function When the original relation and the inverse relation happen to be functions, you can write the inverse of the function f(x) as f–1(x). The two functions are called inverse functions. This notation does not indicate a reciprocal. Functions that undo each other are inverse functions.

Inverse Function

Verify Inverse Functions The inverse function is an “inverse” with respect to the operation of composition of functions. The inverse function “undoes” the function, that is, f -1(f (x)) = x. The function is the inverse of its inverse function, that is, f (f -1(x)) = x. Therefore, to verify that functions are inverses, show that f -1(f (x)) = x and that f (f -1(x)) = x.

EXAMPLE 2 Verify that functions are inverses Verify that f(x) = 3x – 5 and f –1(x) = 1 3 x + 5 are inverse functions. SOLUTION STEP 1 STEP 2 Show: that f(f –1(x)) = x. Show: that f –1(f(x)) = x. f (f –1(x)) = f 3 1 x + 5 f –1(f(x)) = f –1(3x – 5) 3 1 x + 5 = 3 – 5 = 1 3 5 (3x – 5) + = x – 5 3 + = x + 5 – 5 = x = x

Your Turn: for Examples 1 and 2 Find the inverse of the given function. Then verify that your result and the original function are inverses. 1. f(x) = x + 4 ANSWER f –1 (x) = x − 4 f(f –1 (x)) = (x − 4) + 4 = x f –1 (f(x)) = (x + 4) − 4 = x

Your Turn: for Examples 1 and 2 2. f(x) = 2x – 1 ANSWER x + (2) 2 f –1 (x) = f –1 (f(x)) = = = x (2x −1) + 1 2x f(f –1 (x)) = 2 – 1 = x + 1 – 1 = x x + 1

Your Turn: for Examples 1 and 2 3. f(x) = –3x – 1 ANSWER x – 1 –3 f –1 (x) = f –1 (f(x)) = = = x (−3x +1) − 1 −3x −3 f(f –1 (x)) = –3 + 1 = x – 1 + 1 = x

EXAMPLE 3 Solve a multi-step problem Fitness 3 8 Elastic bands can be used in exercising to provide a range of resistance. A band’s resistance R (in pounds) can be modeled by R = L – 5 where L is the total length of the stretched band (in inches).

Solve a multi-step problem EXAMPLE 3 Solve a multi-step problem • Find the inverse of the model. Use the inverse function to find the length at which the band provides 19 pounds of resistance. • SOLUTION STEP 1 Find the inverse function. R = L – 5 3 8 Write original model. R + 5 = 3 8 L Add 5 to each side. 8 3 40 R + = L Multiply each side by 8 3 .

EXAMPLE 3 Solve a multi-step problem STEP 2 Evaluate the inverse function when R = 19. 40 3 L = 8 R + = (19) + 40 3 8 40 3 152 = + 192 3 = = 64 ANSWER The band provides 19 pounds of resistance when it is stretched to 64 inches.

Your Turn: for Example 3 4. Fitness: Use the inverse function in Example 3 to find the length at which the band provides 13 pounds of resistance. ANSWER The band provides 13 pounds of resistance when it is stretched to 48 inches.

Inverses of Nonlinear Functions The graphs of the power functions f(x) = x2 and g(x) = x3 are shown below along with their reflections in the line y = x. Notice that the inverse of g(x) = x3 is a function, passes the vertical line test, but that the inverse of f(x) = x2 is not a function, does not pass the vertical line test.

Inverses of Nonlinear Functions If the domain of f(x) = x2 is restricted to only nonnegative real numbers, then the inverse of f is a function. When the inverse of a function is not a function, the domain of the function can be restricted to allow the inverse to be a function. In such cases, it is convenient to consider “part” of the function by restricting the domain of f(x). If the domain is restricted, then its inverse is a function.

Horizontal Line Test You can use the graph of a function f to determine whether the inverse of f is a function by applying the horizontal line test. If a horizontal line intersects the graph of an equation more than one time, the equation graphed inverse is NOT a function. Inverse is not a Function Inverse is not a Function Inverse is a Function

Horizontal Line Test

Find the inverse of a power function EXAMPLE 4 Find the inverse of a power function Find the inverse of f(x) = x2, x ≥ 0. Then graph f and f –1. SOLUTION f(x) = x2 Write original function. y = x2 Replace f (x) with y. x = y2 Switch x and y. ± = x y Take square roots of each side.

EXAMPLE 4 Find the inverse of a power function The domain of f is restricted to nonnegative values of x. So, the range of f –1 must also be restricted to nonnegative values, and therefore the inverse is f –1(x) = x. (If the domain was restricted to x ≤ 0, you would choose f –1 (x) = – x.)

EXAMPLE 5 Find the inverse of a cubic function Consider the function f (x) = 2x3 + 1. Determine whether the inverse of f is a function. Then find the inverse. SOLUTION Graph the function f. Notice that no horizontal line intersects the graph more than once. So, the inverse of f is itself a function. To find an equation for f –1, complete the following steps:

Find the inverse of a cubic function EXAMPLE 5 Find the inverse of a cubic function f (x) = 2x3 + 1 Write original function. y = 2x3 + 1 Replace f (x) with y. x = 2y3 + 1 Switch x and y. x – 1 = 2y3 Subtract 1 from each side. x – 1 2 = y3 Divide each side by 2. x – 1 2 = y 3 Take cube root of each side.

EXAMPLE 5 Find the inverse of a cubic function ANSWER The inverse of f is f –1(x) = . x – 1 2 3

GUIDED PRACTICE Your Turn: for Examples 4 and 5 Find the inverse of the function. Then graph the function and its inverse. 5. f(x) = x6, x ≥ 0 ANSWER f –1(x) = 6√ x

GUIDED PRACTICE Your Turn: for Examples 4 and 5 6. g(x) = x3 1 27 ANSWER g–1(x) = 33√ x

GUIDED PRACTICE Your Turn: for Examples 4 and 5 7. f(x) = – x3 64 125 ANSWER f –1(x) = – 3√ x 5 4

GUIDED PRACTICE Your Turn: for Examples 4 and 5 8. f(x) = –x3 + 4 ANSWER f –1(x) = 3√ 4 – x

√ GUIDED PRACTICE Your Turn: for Examples 4 and 5 9. f(x) = 2x5 + 3 ANSWER 5

√ GUIDED PRACTICE Your Turn: for Examples 4 and 5 10. g(x) = –7x5 + 7 ANSWER g –1 (x) = √ x – 7 –7 5

Real-World Problem Inverses Anytime you need to undo an operation or work backward from a result to the original input, you can apply inverse functions. In a real-world situation, don’t switch the variables, because they are named for specific quantities. Remember!

EXAMPLE 6 Find the inverse of a power model Ticket Prices The average price P (in dollars) for a National Football League ticket can be modeled by P = 35t0.192 where t is the number of years since 1995. Find the inverse model that gives time as a function of the average ticket price.

Find the inverse of a power model EXAMPLE 6 Find the inverse of a power model SOLUTION P = 35t0.192 Write original model. = t0.192 p 35 Divide each side by 35. = (t0.192)1/0.192 p 35 1/0.192 Raise each side to the power . 1 0.192 p 35 5.2 t Simplify. This is the inverse model.

Use an inverse power model to make a prediction EXAMPLE 7 Use an inverse power model to make a prediction Use the inverse power model from Example 6 to predict the year when the average ticket price will reach $58. SOLUTION t = P 35 5.2 Write inverse power model. = 58 35 5.2 Substitute 58 for P. ≈ 14 Use a calculator.

EXAMPLE 7 Use an inverse power model to make a prediction ANSWER You can predict that the average ticket price will reach $58 about 14 years after 1995, or in 2009.

Your Turn: for Examples 6 and 7 11. Ticket Prices: The average price P (in dollars) for a Major League Baseball ticket can be modeled by P = 10.7t0.272 where t is the number of years since 1995. Write the inverse model. Then use the inverse to predict the year when the average ticket price will reach $25. ANSWER t = P 10.7 3.68 You can predict that the average ticket price will reach $25 about 23 years after 1995, or in 2018.