Question

Show that \(f\) and \(g\) are inverse functions by (a) using the definition of inverse functions and (b) graphing the functions. Make sure you test a few points, as shown in Examples 6 and 7 . \(f(x)=x^{3}, \quad g(x)=\sqrt[3]{x}\)

Short answer

Considering the definition of inverse functions, it has been shown that \(f\) and \(g\) are inverses as both \(f(g(x))\) and \(g(f(x))\) return \(x\). The graph shows that they are mirror images across the line \(y = x\), and the tested points uphold these findings. In conclusion, \(f(x) = x^3\) and \(g(x) = \sqrt[3]{x}\) are indeed inverse functions.

Step-by-step solution

Using the Definition of Inverse Functions

To show that two functions are inverses of each other, the function compositions \(f(g(x))\) and \(g(f(x))\) should both return \(x\). \[ f(g(x)) = f(\sqrt[3]{x}) = (\sqrt[3]{x})^3 = x \] And, \[ g(f(x)) = g(x^3) = \sqrt[3]{x^3} = x \] Hence, it is proven that \(f\) and \(g\) are inverses of each other based on the definition of inverse functions.

Graphing the functions

The graphs of \(f(x) = x^3\) and \(g(x) = \sqrt[3]{x}\) should be reflected across the line \(y = x\) if they are indeed inverse functions. Plot the functions on a graph and you will notice that they are indeed mirror images of each other across the line \(y = x\).

Testing Points

Select a few points and test them. For example, take a point (a, b) on the graph of \(f\). This corresponds to the point \(f(a) = b\). The same point on \(g\) should be \((b, a)\) because \(g(b) = a\). If you carry this out, you will find that the points do indeed match up, further proving that the two functions are inverses.

Key concepts

Function Composition

Function composition is an important concept in mathematics that helps us understand how different functions interact with each other. When you "compose" functions, it means you are plugging the output of one function into the input of another. This concept is critical for verifying inverse functions.

For the functions given in the exercise, we have:

• The function composition \( f(g(x)) \) where \( f(x) = x^3 \) and \( g(x) = \sqrt[3]{x} \).
• Calculating \( f(g(x)) \), we replace \( g(x) \) with \( \sqrt[3]{x} \) in \( f \), giving us \( f(g(x)) = (\sqrt[3]{x})^3 \), which simplifies to \( x \).
• Similarly, the composition \( g(f(x)) \) means plugging \( f(x) = x^3 \) into \( g \), resulting in \( g(f(x)) = \sqrt[3]{x^3} \), which again returns \( x \).

These computations show that when each function is applied to the other's output, we end up back at our starting value \( x \). This property is precisely what defines inverse functions when considered by function composition.

Graphing Functions

Graphing functions is another way to visually illustrate the relationship between functions and their inverses. For the functions \( f(x) = x^3 \) and \( g(x) = \sqrt[3]{x} \), their graphs give us insightful visual confirmation of their inverse relationship.

Here's how you can visually check:

• Plot \( f(x) = x^3 \) and you'll see a curve that passes through the origin (0,0) and extends through all four quadrants as it increases and decreases.
• Next, graph \( g(x) = \sqrt[3]{x} \). This curve should look like the reflection of \( f(x) \) across the line \( y = x \).
• The key reflection line \( y = x \) is an important identity line because if the graphs are true inverses, they "mirror" each other precisely across this line.

By graphing these functions, we can clearly see that the graphs of \( f \) and \( g \) are perfect reflections, which further confirms that they are inverse functions.

Inverse Function Theorem

The Inverse Function Theorem provides mathematical conditions under which a function has an inverse. It generally applies to functions that are continuously differentiable, but it gives us a deeper understanding of how inverses work.

While the theorem is more advanced, the basic idea is that if a function is differentiable and its derivative is non-zero at a point, then a local inverse exists around that point. However, for our example with \( f(x) = x^3 \) and \( g(x) = \sqrt[3]{x} \), we do not need to apply this theorem directly, but understanding it helps reinforce why these functions can have inverses.

• The function \( f(x) = x^3 \) is continuously differentiable and its derivative \( f'(x) = 3x^2 \) is non-zero for all \( x eq 0 \).
• This indicates that \( f \) is locally invertible around any point except \( x = 0 \), though even at \( x = 0 \), the global structure of cubic and cube root transformations allows for complete inverse functions.

Hence, the Inverse Function Theorem underpins the smooth and well-behaved nature of these functions' inversion process across their entire domain.