Problem 199 - Calculus Volume 1

Question

For the following exercises, use composition to determine which pairs of functions are inverses. $$ f(x)=8 x, g(x)=\frac{x}{8} $$

Step-by-Step Solution

Answer

Functions \(f(x) = 8x\) and \(g(x) = \frac{x}{8}\) are inverses.

Step 1: Understand the Definition of Inverse Functions

Two functions, \(f(x)\) and \(g(x)\), are inverses of each other if the composition of \(f\) and \(g\) in both orders results in the identity function. In mathematical terms, this means: 1. \( (f \circ g)(x) = x \) for all \(x\) in the domain of \(g\).2. \( (g \circ f)(x) = x \) for all \(x\) in the domain of \(f\). This means applying \(f\) then \(g\), or vice versa, returns the original value of \(x\).

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

To compute \((f \circ g)(x)\), substitute \(g(x)\) into \(f\): \[(f \circ g)(x) = f(g(x)) = f\left(\frac{x}{8}\right) = 8 \times \frac{x}{8} = x.\] This simplifies to \(x\), which shows that part of the criteria for inverse functions is satisfied.

Step 3: Compute \((g \circ f)(x)\)

To compute \((g \circ f)(x)\), substitute \(f(x)\) into \(g\): \[(g \circ f)(x) = g(f(x)) = g(8x) = \frac{8x}{8} = x.\] This also simplifies to \(x\), thus fulfilling the other part of the criteria for inverse functions.

Step 4: Conclude Using the Defined Conditions

Since both \((f \circ g)(x) = x\) and \((g \circ f)(x) = x\) hold true for all \(x\), functions \(f(x) = 8x\) and \(g(x) = \frac{x}{8}\) are inverses of each other.

Key Concepts

Function Composition

Function composition is essentially putting one function inside another. Specifically, we want to see what happens when we use the output of one function as the input for another. In mathematical terms, composition is denoted as

• For two functions, say \(f(x)\) and \(g(x)\), composing \(f\) and \(g\) is expressed as \((f \circ g)(x)\), meaning \(f(g(x))\).
• Similarly, \((g \circ f)(x)\) would mean \(g(f(x))\).

The key goal of composition, especially in the context of inverse functions, is to see if the composition results in the identity function. In simple terms, you want to find out if plugging one function into another just returns the input variable \(x\). For example, when we say \((f \circ g)(x) = x\), we are checking if applying \(g(x)\) first and then \(f(x)\) brings us back to \(x\), confirming the functions are indeed inverses.

Identity Function

The identity function is a straightforward concept. It's a function that always returns the input as its output. So, for a function \(f(x)\) to be the identity function, it should satisfy:

• \( f(x) = x \) for every \(x\) in its domain.

When discussing inverse functions, the identity function is crucial. It serves as the gold standard for checking if two functions are inverses of each other.When the composition of two functions, \(f\) and \(g\), results in the identity function

• \((f \circ g)(x) = x\) and
• \((g \circ f)(x) = x\)

for their respective domains, they are considered as inverse functions. The identity function tells you that taking one step forward and one step back from any point leaves you exactly where you started.

Domain of a Function

The domain of a function is the set of all possible input values \(x\) for which the function is defined. It's like the function's playground where it can "play" or "act".When you're dealing with function compositions, especially checking for inverses, domains become very important.

• If two functions, \(f\) and \(g\), are true inverses, then their composition must return the identity function within the respective domains.
• For instance, \((f \circ g)(x) = x\) needs to hold for all \(x\) in the domain of \(g\), and \((g \circ f)(x) = x\) must hold for all \(x\) in the domain of \(f\).

Understanding domains helps ensure that the functions work correctly across all intended inputs. It's important to verify that the "inverse" relationship holds everywhere it's supposed to, not just for specific or limited inputs.