Problem 15 - Precalculus : Building Concepts and Connections

Question

Verify that the given functions are inverses of each other. $$f(x)=x^{3}+2 ; g(x)=\sqrt[3]{x-2}$$

Step-by-Step Solution

Answer

The functions \(f(x)=x^{3}+2\) and \(g(x)=\sqrt[3]{x-2}\) are indeed inverses of each other.

Step 1: Compute \(f(g(x))\)

Here, we substitute \(g(x) = \sqrt[3]{x-2}\) into the function \(f(x) = x^{3}+2\): \(f(g(x)) = (\sqrt[3]{x-2})^{3}+2\) . After simplification we obtain: \(f(g(x)) = x\).

Step 2: Compute \(g(f(x))\)

For this, we substitute \(f(x) = x^{3}+2\) into the function \(g(x) = \sqrt[3]{x-2}\): \(g(f(x)) = \sqrt[3]{(x^{3}+2)-2}\). This simplifies into: \(g(f(x)) = \sqrt[3]{x^{3}}\), which simplifies further into \(g(f(x)) = x\).

Step 3: Conclusion

Since we got \(f(g(x)) = x\) and \(g(f(x)) = x\), we can say that \(f\) and \(g\) are indeed inverses of each other.

Key Concepts

Precalculus and Inverse Functions

Precalculus is a foundational course that prepares students for studying calculus. One of the crucial concepts in precalculus is that of inverse functions. An inverse function essentially reverses the operation of a given function. For example, if you have a function that adds 2 to any input value, its inverse would subtract 2 from any input value.

To verify that two functions are inverses of each other, we check whether composing one with the other brings us back to our original input value. Specifically, if we have functions f and g, we say that g is the inverse of f if, for every x in the domain of f, the equation f(g(x)) = x holds true. Similarly, g(f(x)) must also equal x for every x in the domain of g. This test ensures that the functions 'undo' each other.

Composite Functions

Composite functions come into play when we talk about inverses because they involve the application of one function to the results of another. We denote a composite function by f(g(x)), which means that we first apply g to x and then apply f to the result of g(x).

To verify inverses, as shown in the original exercise, we compute the composite of f and g in both orders: f(g(x)) and g(f(x)). If both composite functions simplify to x, then f and g are confirmed to be inverses of each other. Composite functions are particularly useful for this purpose, as they show the processes of the original function, and its inverse, directly undoing each other.

Function Simplification

When working with functions, simplification is often necessary to see the underlying connections, like identifying inverses. The goal of simplification is to make functions more manageable and to express them in a simpler form.

In the context of the inverse function exercise, simplification is used to show that f(g(x)) and g(f(x)) both boil down to x. This involves algebraic manipulations such as expanding, factoring, and canceling terms where appropriate. Simplifying complex expressions helps in identifying crucial features of functions, such as their behavior, intercepts, and, of course, their inverses.

Cubic Functions

Cubic functions are polynomial functions of degree three, and they typically have the form f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants, and a is non-zero. They are known for their distinctive 'S' shaped curve when graphed.

Cubic functions can be inverted by finding their cube root counterparts, as illustrated in our exercise where the cubic function f(x) = x^3 + 2 has an inverse g(x) = \(\sqrt[3]{x-2}\). Working with the inverses of cubic functions often involves the use of cube roots. Finding the inverse of a cubic function can be more algebraically intensive than linear or quadratic functions, since it requires understanding the behavior of the cube and cube root operations.