Question

Find the inverse function of \(f\). Use a graphing utility to graph \(f\) and \(f^{-1}\) in the same viewing window. Describe the relationship between the graphs. $$ f(x)=3 \sqrt[5]{2 x-1} $$

Short answer

The inverse function of \(f(x) = 3 \sqrt[5]{2x-1}\) is \(f^{-1}(x) = 0.5x^5/243 + 0.5. When graphed, this inverse function is a reflection of the original function over the line \(y = x\), showing the inherent symmetry between a function and its inverse.

Step-by-step solution

Write down the function and replace \(f(x)\) with \(y\)

Start with the function \(f(x)\). To make it easier to work with, let's replace \(f(x)\) with \(y\). So, we write down \(y = 3 \sqrt[5]{2 x-1}\)

Swap \(x\) and \(y\)

Now, swap the roles of \(x\) and \(y\). This will give us \(x = 3 \sqrt[5]{2 y-1}\)

Solve the equation for \(y\)

Now, solve the equation for \(y\) which will be our inverse function \(f^{-1}(x)\). First, divide both sides by 3, yielding \(x/3 = \sqrt[5]{2 y-1}\). Then, to eliminate the 5th root, raise both sides to the power 5: \(x^5/243 = 2 y - 1\). Finally, add 1 and divide by 2 to isolate \(y\), giving the inverse function \(f^{-1}(x) = 0.5x^5/243 + 0.5\)

Graph \(f(x)\) and \(f^{-1}(x)\)

Use a graphing utility to plot both \(f(x) = 3 \sqrt[5]{2 x-1}\) and \(f^{-1}(x) = 0.5x^5/243 + 0.5\) on the same graph.

Analyze the relationship between the graphs

Both functions will be symmetrical to the line \(y=x\). This occurs because the inverse of a function is a reflection of the original function over the line \(y = x\) This will show the direct relationship between \(f\) and \(f^{-1}\).

Key concepts

Graphing Inverse Functions

When graphing inverse functions, understanding the relationship between a function and its inverse is critical. Take the function given in the exercise,
\(f(x)=3 \sqrt[5]{2x-1}\)
. To graph its inverse, first, understand that the inverse function, denoted by \(f^{-1}(x)\), will undo the action of \(f(x)\). After obtaining the inverse function, which in this case is
\(f^{-1}(x) = 0.5x^5/243 + 0.5\),
it's time to graph both functions.

Procedure for Graphing

• Start by graphing the original function \(f(x)\). Plot several points on the coordinate plane by assigning different x-values and solving for y.
• Next, graph the inverse function \(f^{-1}(x)\) by similarly plotting points, but keep in mind that the roles of x and y are swapped in comparison to \(f(x)\).
• Use a graphing utility if available, as it allows for a more accurate representation, especially with functions involving roots and higher powers.

After plotting, you will notice the symmetry, which indicates that the graphing process has been done correctly. This visual understanding is essential for comprehending the interplay between functions and their inverses.

Solving Inverse Functions

To solve inverse functions such as \(f^{-1}(x)\) for the given function \(f(x)=3 \sqrt[5]{2 x-1}\), one must follow a systematic procedure to 'undo' the operations performed by the original function. The exercise provided shows the steps involved in solving for the inverse.

Steps to Solve for Inverse Functions

• Step 1: Write the function as \(y\) to prepare for inverting. Here, we start with \(y = f(x)\).
• Step 2: Swap the variables \(x\) and \(y\) to obtain a new equation, where the former outputs become inputs and vice versa.
• Step 3: Solve the resulting equation for \(y\), which will now represent \(f^{-1}(x)\). This might involve algebraic manipulation, such as isolating the variable through activities like factoring, distributing, or applying inverse operations.

Following these steps ensures one can find the inverse function algebraically. Understanding how to perform these steps is crucial for tackling a wide range of problems involving inverse functions in calculus.

Symmetry in Inverse Functions

Symmetry in inverse functions plays a significant role in visually understanding these mathematical entities. The symmetry is always with respect to the line \(y=x\). This concept is elegantly illustrated in the step-by-step solution where the functions \(f(x)\) and \(f^{-1}(x)\) are graphed.

When graphing the original function and its inverse on the same Cartesian plane, one observes that every point on the graph of the original function has a corresponding point on the graph of its inverse that is reflected across the line \(y=x\).

Importance of Symmetry

• Symmetry helps to confirm that the function you've derived as the inverse is correct; if the graph doesn't exhibit this symmetry, then the inverse function may have been found incorrectly.
• This reflection property also aids in understanding how the outputs of \(f(x)\) relate to the inputs of \(f^{-1}(x)\), and vice versa.
• It provides a visual way to check the work: if the graph of the original function is known, the graph of the inverse can be estimated through reflection.

The concept of symmetry is more than a visual aid; it's a fundamental property of inverse functions that serves as a critical check for correctness in analytical work.