Problem 15 - College Algebra with Corequisite Support

Question

For the following exercises, use each pair of functions to find \(f(g(x))\) and \(g(f(x))\). Simplify your answers. \(f(x)=\sqrt[3]{x}, \quad g(x)=\frac{x+1}{x^{3}}\)

Step-by-Step Solution

Answer

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

Step 1: Defining f(g(x))

The task is to find the composition of the functions where the output of function \(g(x)\) is the input for function \(f(x)\). So, we will substitute \(g(x)\) into \(f(x)\).

Step 2: Substituting g(x) in f(x)

Since \(f(x) = \sqrt[3]{x}\), replace \(x\) with \(g(x)\): \(f(g(x)) = \sqrt[3]{\frac{x+1}{x^{3}}}\).

Step 3: Defining g(f(x))

Next, find the composition \(g(f(x))\), where the output of \(f(x)\) is the input for \(g(x)\). This means we substitute \(f(x)\) into \(g(x)\).

Step 4: Substituting f(x) in g(x)

Since \(g(x) = \frac{x+1}{x^3}\), replace \(x\) with \(f(x) = \sqrt[3]{x}\): \(g(f(x)) = \frac{\sqrt[3]{x} + 1}{(\sqrt[3]{x})^3}\).

Step 5: Simplifying g(f(x))

Simplify \(g(f(x)) = \frac{\sqrt[3]{x} + 1}{x}\). Since \((\sqrt[3]{x})^3 = x\), the expression simplifies to \(\frac{\sqrt[3]{x} + 1}{x}\).

Key Concepts

Inverse Functions

Inverse functions are essential in mathematics because they help us "undo" a function, much like how division undoes multiplication. When dealing with a function, an inverse function is one that reverses the effect of the original function. In simpler terms, if you apply a function and then its inverse, you should arrive back at your starting value.

• The notation for an inverse function is usually written as \(f^{-1}(x)\). This doesn't mean \(1/f(x)\) but instead refers to the inverse operation.
• For instance, if \(f(x) = x^3\), then \(f^{-1}(x) = \sqrt[3]{x}\), because applying \(x^3\) to a number and then taking the cube root brings you back to the original number.
• However, not all functions have inverses. A function must be one-to-one (bijective), meaning each output is linked to one unique input.

Understanding inverse functions is crucial when working with function compositions, like in our exercise with \(f(x) = \sqrt[3]{x}\) and \(g(x) = \frac{x+1}{x^3}\). It helps determine if the operations can be reversed.

Cube Roots

Cube roots are relatively straightforward; they are the inverse of cubing a number. The cube root of a number \(x\) is a number \(y\) such that \(y^3 = x\). In mathematical terms, this is expressed as \(\sqrt[3]{x}\).

• This operation is significant because it undoes the cubing operation. For instance, if you have \(x = 8\), then \(\sqrt[3]{x} = 2\), because \(2^3 = 8\).
• Cube roots can be applied not only to positive numbers but also to negative ones. For example, the cube root of \(-27\) is \(-3\), since \((-3)^3 = -27\).
• In our context, \(f(x) = \sqrt[3]{x}\) means we are taking the cube root of whatever is substituted in for \(x\), which in function composition becomes a vital step for transformations and simplifying calculations.

By understanding cube roots, you can manipulate and simplify expressions, crucial when working with complex function compositions.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. They're similar to rational numbers, but instead of simple numbers, you have "polynomial numbers" in the fraction form.

• A typical rational expression looks like \(\frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials.
• One must be careful with rational expressions because the denominator must never be zero. Values that make the denominator zero are restricted from the domain.
• In our exercise with \(g(x) = \frac{x+1}{x^3}\), the denominator \(x^3\) means \(x\) cannot be zero. Otherwise, the expression becomes undefined.
• When simplifying rational expressions, look for common factors in the numerator and denominator that can be cancelled out to simplify calculations.

Mastering rational expressions is key to working with function compositions and understanding how to manipulate them for clearer, more concise mathematical solutions.