Question

In Exercises 39-52, a. Find an equation for ƒ¯¹(x). b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range off and ƒ¯¹. f(x) = x³ − 1

Accepted Answer

Step 1: To find the inverse function ƒ¯¹(x), start by replacing f(x) with y. This gives y = x³ - 1. Then, swap x and y to reflect the inverse relationship. The equation becomes x = y³ - 1. Step 2: Solve for y in terms of x to isolate the inverse function. Add 1 to both sides of the equation to get x + 1 = y³. Then, take the cube root of both sides to solve for y, resulting in y = ∛(x + 1). Thus, the inverse function is ƒ¯¹(x) = ∛(x + 1). Step 3: To graph both f(x) = x³ - 1 and ƒ¯¹(x) = ∛(x + 1) on the same rectangular coordinate system, plot several points for each function. For f(x), choose x-values, compute f(x), and plot the points. For ƒ¯¹(x), choose x-values, compute ƒ¯¹(x), and plot the points. Remember that the graphs of f and ƒ¯¹(x) are symmetric about the line y = x. Step 4: Determine the domain and range of f(x) = x³ - 1. Since the cube function x³ is defined for all real numbers, the domain of f(x) is (-∞, ∞). The range of f(x) is also (-∞, ∞) because the cube function can produce all real numbers. Step 5: Determine the domain and range of ƒ¯¹(x) = ∛(x + 1). The cube root function ∛(x) is defined for all real numbers, so the domain of ƒ¯¹(x) is (-∞, ∞). The range of ƒ¯¹(x) is also (-∞, ∞) because the cube root function can produce all real numbers.

In Exercises 39-52, a. Find an equation for ƒ¯¹(x). b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range off and ƒ¯¹. f(x) = x³ − 1

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