91Ó°ÊÓ

Inverse functions are pairs of functions that work as opposites of each other. When you apply one function and then the other, you end up where you started. In other words, if you have a function \(f\) and an inverse function \(g\), then \(f(g(x)) = x\) and \(g(f(x)) = x\).
For the given functions, \(f(x) = 4x\) and \(g(x) = \frac{1}{4}x\), it's clear that \(g(x)\) is the inverse of \(f(x)\).
You can think of \(f(x)\) as multiplying \(x\) by 4, and \(g(x)\) as dividing \(x\) by 4. These operations undo each other, making \(f\) and \(g\) inverse functions.

Function operations include addition, subtraction, multiplication, division, and composition of functions. Here, we focus on function composition. Composing functions means applying one function to the result of another function. For functions \(f\) and \(g\), the composition is noted as \(f \big(g(x)\big)\). Let's look at the given exercise:

• \(f \big(g(x)\big)\) is applying \(f\) to \(g(x)\)
• \(g \big(f(x)\big)\) is applying \(g\) to \(f(x)\)

For \(f(x) = 4x\) and \(g(x) = \frac{1}{4}x\), the composition operations are straightforward. We start with \(g(x)\) and then apply \(f\) to it: \(f \big(g(x)\big) = f \big(\frac{1}{4}x\big) = 4 \times \frac{1}{4}x = x\). Similarly, we start with \(f(x)\) and then apply \(g\) to it: \(g \big(f(x)\big) = g (4x) = \frac{1}{4} \times 4x = x\).

Algebraic proofs involve showing that an equation or statement holds true using algebraic methods and reasoning. For the given exercise, we need to show that \((f \big(g(x)\big) = x\) and \((g \big(f(x)\big) = x\). This involves:

• Substituting one function into another
• Simplifying the result to see if it equals to \(x\)

1. For \(f \big(g(x)\big)\), we simplify it as follows: \(f \big(g(x)\big) = f\big(\frac{1}{4}x\big) = 4 \frac{1}{4}x = x\). This shows that \(f \big(g(x)\big) = x\).
2. For \(g \big(f(x)\big)\), we do the similar simplification: \(g \big(f(x)\big) = g \big(4x\big) = \frac{1}{4}4x = x\). This proves that \(g \big(f(x)\big) = x\).
Hence, we have proven algebraically that \((f \big(g(x)\big) = x\) and \((g \big(f(x)\big) = x\) when \(f(x) = 4x\) and \(g(x) = \frac{1}{4}x\).