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+2)³

Verified step by step guidance

1. Step 1: To find the inverse function ƒ¯¹(x), start by replacing f(x) with y. This gives y = (x + 2)³. Then, swap x and y to begin solving for the inverse. The equation becomes x = (y + 2)³.
2. Step 2: Solve for y in terms of x to find the inverse function. Take the cube root of both sides to isolate (y + 2). This gives ³√x = y + 2. Finally, subtract 2 from both sides to solve for y, resulting in ƒ¯¹(x) = ³√x - 2.
3. Step 3: To graph ƒ(x) = (x + 2)³ and ƒ¯¹(x) = ³√x - 2 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.
4. Step 4: Determine the domain and range of ƒ(x). Since the function (x + 2)³ is a cubic function, it is defined for all real numbers. Thus, the domain of ƒ(x) is (-∞, ∞). The range of a cubic function is also all real numbers, so the range of ƒ(x) is (-∞, ∞).
5. Step 5: Determine the domain and range of ƒ¯¹(x). The inverse function ƒ¯¹(x) = ³√x - 2 is a cube root function, which is also defined for all real numbers. Therefore, the domain of ƒ¯¹(x) is (-∞, ∞), and its range is (-∞, ∞).

Key Concepts

Inverse Functions

An inverse function reverses the effect of the original function. If f(x) transforms x into y, then the inverse function f¯¹(y) transforms y back into x. To find the inverse, we typically swap the roles of x and y in the equation and solve for y. Understanding this concept is crucial for part (a) of the question, where we need to derive the inverse of the given function.

Graphing Functions

Graphing functions involves plotting points on a coordinate system to visualize the relationship between the input (x) and output (f(x)). For the original function f(x) and its inverse f¯¹(x), their graphs will reflect across the line y = x. This symmetry is important for part (b) of the question, as it helps in accurately representing both functions on the same graph.

Domain and Range

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (f(x)). For the original function and its inverse, the domain of one is the range of the other and vice versa. Understanding how to express these sets in interval notation is essential for part (c) of the question, as it requires identifying and stating the domain and range for both functions.