Question

In Exercises 35–46, determine whether the inverse of \(f\) is a function. Then find the inverse. $$ f(x)=\frac{1}{2} x^5 $$

Short answer

The inverse of \(f(x) = \frac{1}{2} x^5\) is \(f^{-1}(x) = (2x)^{1/5}\), and it is a function since it passes the vertical line test.

Step-by-step solution

Swap x and y and Solve for y

To find the inverse of a function, we start by replacing the function of \(f(x)\) with \(y\). So, \(y = \frac{1}{2} x^5\). It implies \(x = \frac{1}{2} y^5\). Now, we want to solve for \(y\). So, we have \(2x = y^5\). This gives \(y = (2x)^{1/5}\). Hence, \(f^{-1}(x) = (2x)^{1/5}\)

Verify if the Inverse is a Function

To verify if the inverse we have obtained is a function, we can apply the vertical line test on \(f^{-1}(x) = (2x)^{1/5}\). The test stipulates that when a vertical line cuts the graph at more than one point, then the graph does not represent a function. Since the fifth root function is a function, its output will pass the vertical line test, thus the inverse of \(f(x) = \frac{1}{2} x^5\) is a function.

Key concepts

Function Verification

When working on inverse functions, the first step is crucial: verifying whether the inverse you found is indeed a function. In our problem, the original function is given by \(f(x) = \frac{1}{2} x^5\). We aim to determine if its inverse is a function.

The verification process starts with expressing the inverse. Once we rearrange the equation and solve for \(y\), we establish \(f^{-1}(x) = (2x)^{1/5}\). However, simply finding this inverse isn't enough.

We need to apply specific tests, such as the vertical line test (which we will cover later) to ensure the output is actually a function. This involves verifying that the inverse operation produces a single, unique output for every input in the domain, maintaining the essential characteristic of a function.

• The process of finding the inverse involved swapping \(x\) and \(y\) in \(y = \frac{1}{2} x^5\), resulting in \(y = (2x)^{1/5}\).
• This rearrangement is a core aspect of the verification process.
• Ensure that the operations provide valid and consistent output across the evaluated range.

Thus, by verifying, we confirm that the inverse function is legitimate and can be used reliably for further analysis.

Function Graph

Visual representation is a powerful approach to understanding inverse functions. Graphing can provide clarity on behaviors and properties of functions.

When you graph an inverse function like \(f^{-1}(x) = (2x)^{1/5}\), it becomes easier to visually inspect it. For instance, you observe if the curve consistently provides a single output for every input value.

To visualize the function graph:

• Plot enough points by choosing specific \(x\) values and computing their corresponding \(y\) values.
• You might note that the inverse function \(f^{-1}(x)\) will be a reflection over the line \(y = x\) when compared to the original function \(f(x)\).
• This characteristic is rooted in the foundation of inverse functions.

Graphing serves as a visual tool which aids comprehension, confirming calculations and theoretical understanding made during function verification.

Vertical Line Test

The vertical line test is an essential graphical technique that helps us determine if a relation is a function. It's particularly advantageous when evaluating inverse functions.

Apply this test by drawing vertical lines at various points along the x-axis on your graph of the function \(f^{-1}(x) = (2x)^{1/5}\). If a vertical line touches the graph of the function at more than one point, it indicates that the graph does not represent a function.

For \(f^{-1}(x)\), each vertical line should intersect the graph at only one point, indicating it's a function. This is intuitive, as the fifth root function's graphical behavior inherently satisfies this test.

• Draw several vertical lines for comprehensive coverage.
• Observe each intersection point or lack thereof carefully.
• This test reinforces the function's credibility graphically.

The vertical line test works seamlessly together with algebraic processes to ensure a holistic approach to function verification.