Problem 45 - College Algebra Essentials

Question

a. Find an equation for \(f^{-1}(x)\) b. Graph \(f\) and \(f^{-1}\) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range of \(f\) and \(f^{-1}\) $$f(x)=x^{3}-1$$

Step-by-Step Solution

Answer

The inverse function of \(f(x) = x^3 - 1\) is \(f^{-1}(x) = (x + 1)^{1/3}\). Both this function and its inverse can be graphed on the same coordinate system. The domain and range of \(f(x)\) are \((- \infty, \infty)\) and those for \(f^{-1}(x)\) are \([-1, \infty)\) and \((- \infty, \infty)\) respectively.

Step 1: Inverse function

First, replace \(f(x)\) with \(y\). So, \(y = x^3 - 1\). Swap \(x\) and \(y\) to find the inverse: \(x = y^3 - 1\). Solve this equation for \(y\). Add 1 to both sides:\(x + 1 = y^3\). Then, take the cubed root on both sides: \(y = (x + 1)^{1/3}\).So, the inverse function \(f^{-1}(x) = (x + 1)^{1/3}\).

Step 2: Graph the Functions

Graph the function and its inverse on the same coordinate system. Note that \(f(x)\) is a regular cubic function shifted down one unit and \(f^{-1}(x)\) is a cube root function shifted over one unit to the left. The points \((1, 0 )\) on f(x) and \((0, 1)\) on \(f^{-1}(x)\) should reflect across the line \(y = x\), which is a property of inverse functions.

Step 3: Domain and Range

The domain and range can also be determined. The domain of a function is the set of all possible input values (often \(x\) values) while the range is the set of all possible output values (often \(y\) values).For \(f(x) = x^3 - 1\), \(x\) could be any real number so its domain is \(-\infty, \infty\). Similarly, the output could be any real number; so the range is also \(-\infty, \infty\).For inverse function \(f^{-1}(x) = (x + 1)^{1/3}\), \(x\) could be any number from \(-1\) to \(\infty\) after it's shifted one unit to the left, so its domain is \([-1, \infty)\). The output could be any real number; so the range is also \(-\infty, \infty\).