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) = ∛x + 1

Accepted Answer

Step 1: To find the inverse function ƒ¯¹(x), start by replacing f(x) with y. This gives y = ∛x + 1. Next, swap x and y to reflect the inverse relationship. This results in x = ∛y + 1. Step 2: Solve for y in terms of x. Subtract 1 from both sides to isolate the cube root term: x - 1 = ∛y. Then, cube both sides to eliminate the cube root: (x - $$1)^3 = y. $$Thus, the inverse function is ƒ¯¹(x) = (x - $$1)^3. $$Step 3: To graph f(x) = ∛x + 1 and its inverse ƒ¯¹(x) = (x - $$1)^3 on $$the same coordinate system, note that the graph of an inverse function is a reflection of the original function across the line y = x. Plot several points for both functions and ensure symmetry about the line y = x. Step 4: Determine the domain and range of f(x). The cube root function ∛x is defined for all real numbers, and adding 1 does not restrict the domain. Therefore, the domain of f(x) is (-∞, ∞). The range of f(x) is also (-∞, ∞) because the cube root function can output all real numbers. Step 5: Determine the domain and range of ƒ¯¹(x). Since the inverse function reflects the domain and range of the original function, the domain of ƒ¯¹(x) is (-∞, ∞), and the range of ƒ¯¹(x) is also (-∞, ∞). Use interval notation to express these results.

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) = ∛x + 1

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Inverse Functions

Graphing Functions

Domain and Range