Problem 15 - College Algebra and Calculus: An Applied Approach

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 . \(f(x)=1-x^{3}, \quad g(x)=\sqrt[3]{1-x}\)

Step-by-Step Solution

Answer

The given functions \(f(x) = 1 - x^{3}\) and \(g(x) = \sqrt[3]{1-x}\) are indeed inverses of each other. This can be confirmed by substituting \(g(x)\) into \(f(x)\) and vice versa which results in \(x\) in both cases, as well as visually through graphing where the graph of \(g(x)\) is found to be a reflection of \(f(x)\) over the line \(y=x\).

Step 1: Verify by Definition

To verify that two functions are inverses of each other using the definition involves two steps. First, substitute function \(g(x)\) into \(f(x)\). If you get \(x\) as a result, you proceed to substituting function \(f(x)\) into \(g(x)\). If you again get \(x\) as a result, it proves that both functions are inverse of each other. \n So start with \(f(g(x))\), which is \(f(\sqrt[3]{1-x})\). After doing the necessary substitutions and simplifications, the result should be \(x\). Next, you calculate \(g(f(x))\), which is \(g(1 - x^{3})\). After doing the necessary substitutions and simplifications, the result should again be \(x\). Therefore, this confirms that \(f(x)\) and \(g(x)\) are indeed inverses of each other.

Step 2: Verify by Graph

To confirm that two functions are inverses of each other using graphing, sketch the graphs of both functions. This involves selecting values for \(x\), computing respective \(y\) values for each function, plotting points and then drawing a smooth curve through the points. Once done, you should have two graphical representations, one for function \(f(x)\) and the other for function \(g(x)\). If the function \(g(x)\) is the reflection of the function \(f(x)\) in the line \(y=x\), then the two graphs are inverses of each other. For this exercise, you will indeed find that graphing confirms that \(f(x)\) and \(g(x)\) are inverses of each other.

Key Concepts

Definition of Inverse Functions

Inverse functions are a pair of functions that reverse the effect of each other. Essentially, if you apply one function and then its inverse, you end up back where you started. In mathematical terms, two functions, say f and g, are inverses if f(g(x)) = x and also g(f(x)) = x for every x in the domains of f and g.

In the exercise provided, function f subtracts the cube of x from 1, while function g takes the cube root of 1 minus x. To show these are inverses, you substitute one function into the other, verifying that f(g(x)) simplifies to x, and vice versa for g(f(x)). This check is critical since it confirms that each function undoes the operation of the other.

Remember, not all functions have inverses. For a function to have an inverse, it must be bijective, meaning it's both injective (one-to-one) and surjective (onto).

Graphing Functions

When graphing functions, the goal is to visually represent their behavior.

Choosing Points

Start by selecting a range of x-values and calculating the corresponding y-values. These coordinate pairs can then be plotted on a graph.

For inverse functions, after plotting, you should notice that each function is a mirror image of the other across the line y = x. This line has a 45-degree slope and acts as a reflective surface.

Plotting the Inverses

If the exercise asks to show two functions are inverses using graphing, as in our case, you would graph both functions and check for this reflective symmetry. The intersection or 'kissing' of the two graphs on the line y = x is a visual confirmation of their inverse relationship.

Composite Functions

A composite function involves one function being applied after another. Mathematically, it is denoted as (f ∘ g)(x) = f(g(x)), which means you first apply function g to x, then apply function f to the result of g(x).

To analyze whether two functions are inverses, we effectively check if applying them consecutively returns the original value of x. This was done in the given exercise, showing that f(g(x)) = x and g(f(x)) = x.

Composing functions is a foundational concept, especially when dealing with inverse functions, because it allows us to understand the undoing process inherent in the concept of inverses. When we see that a composition of functions yields the identity function, we have a strong indication of an inverse relationship.