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

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1. <b>Step 1:</b> To find the inverse function $f^{-1}(x)$, start by replacing $f(x)$ with $y$. So, we have $y=(x+2{)}^3$.
2. <b>Step 2:</b> Swap $x$ and $y$ to find the inverse. This gives us $x=(y+2{)}^3$.
3. <b>Step 3:</b> Solve for $y$ in terms of $x$. Start by taking the cube root of both sides: $\sqrt[3]{x}=y+2$.
4. <b>Step 4:</b> Isolate $y$ by subtracting 2 from both sides: $y=\sqrt[3]{x}-2$. Thus, the inverse function is $f^{-1}(x)=\sqrt[3]{x}-2$.
5. <b>Step 5:</b> For the domain and range, note that the domain of $f(x)=(x+2{)}^3$ is all real numbers, and its range is also all real numbers. The inverse function $f^{-1}(x)=\sqrt[3]{x}-2$ also has a domain and range of all real numbers.

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 finding f¯¹(x) in the given problem.

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 key to understanding how the two functions relate to each other and aids in accurately graphing both functions.

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 f(x) and its inverse f¯¹(x), the domain of one is the range of the other and vice versa. Using interval notation to express these sets provides a clear and concise way to communicate the limits of the functions' inputs and outputs.