Problem 12 - Precalculus : Building Concepts and Connections

Question

Verify that the given functions are inverses of each other. $$f(x)=-8 x ; g(x)=-\frac{1}{8} x$$

Step-by-Step Solution

Answer

By substituting \(g(x)\) into \(f(x)\) and \(f(x)\) into \(g(x)\), and simplifying both expressions, it is confirmed that both results equal to \(x\). Therefore, the functions are inverses of each other.

Step 1: Composition of \(f(x)\) with \(g(x)\)

To start, substitute \(g(x)\) into \(f(x)\), this yields: \[f(g(x)) = f\left(-\frac{1}{8}x\right) = -8\left(-\frac{1}{8}x\right).\] Simplify this expression to verify if it yields \(x\).

Step 2: Simplify the Expression

On simplifying we have \[-8\left(-\frac{1}{8}x\right) = x.\] This implies that \(f(g(x)) = x\), which is one of the conditions for \(f\) and \(g\) to be inverse of each other.

Step 3: Composition of \(g(x)\) with \(f(x)\)

Next, substitute \(f(x)\) into \(g(x)\), this results in: \[g(f(x)) = g(-8x) = -\frac{1}{8}(-8x).\] Simplify this as well to verify if it also yields \(x\).

Step 4: Simplify the Expression

Upon simplifying we get \[-\frac{1}{8}(-8x) = x.\] This implies that \(g(f(x)) = x\), which is the other condition for \(f\) and \(g\) to be inverse of each other.

Key Concepts

Understanding Function Composition

Function composition involves combining two functions in such a way that the output of one function becomes the input of the other. When we say we are finding the composition of functions \(f\) and \(g\), we write it as \( f(g(x)) \) or \( g(f(x)) \). This means you take the entire function \(g(x)\) and plug it into every \(x\) in \(f(x)\), or vice versa.

• For example, if \( f(x) = -8x \) and \( g(x) = -\frac{1}{8}x \), then \( f(g(x)) = f\left(-\frac{1}{8}x\right) \).
• This results in substituting \(-\frac{1}{8}x\) into \(f\), making it \( -8(-\frac{1}{8}x) \).

The goal is to simplify this to check if the resulting expression returns our original \(x\). If both \(f(g(x))\) and \(g(f(x))\) simplify to \(x\), the functions are inverses of each other.

The Art of Simplifying Expressions

Simplifying expressions involves reducing a complex equation or expression into its simplest form. It requires removing parentheses, combining like terms, and reducing fractions where possible. When dealing with function composition, each step should simplify every part of the expression resulting from the substitution.
For instance, given the expression \(-8\left(-\frac{1}{8}x\right)\), we simplify by:

• First, multiply the constants: \(-8\) and \(-\frac{1}{8}\).
• This simplifies to \((-8) \times (-\frac{1}{8}) = 1\).
• Therefore, the whole expression reduces to \(1 \times x = x\).

Each simplification step should be straightforward and aim to find whether our resulting expression turns back into \(x\). It is crucial for verifying that our than composed functions are inverses.

Conditions for Inverse Functions

For two functions to be considered inverses, specific conditions must be met. The primary condition is that the result of function composition both ways, \(f(g(x))\) and \(g(f(x))\), must equal \(x\).

• This means if you put any value into \(g(x)\) and then into \(f(x)\), or vice versa, the output should be the initial input \(x\).
• In our example, substituting \(g(x)\) into \(f(x)\) gave \(-8(-\frac{1}{8}x) = x\).
• Similarly, \(g(f(x))\) = \(-\frac{1}{8}(-8x)\) also simplifies to \(x\).

These compositions prove that \(f\) and \(g\) are truly inverses, serving as a thorough check to ensure they undo each other’s operations. This foundational understanding is critical when verifying inverse functions in any exercise.