Problem 1 - Precalculus Essentials

Question

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=4 x \text { and } g(x)=\frac{x}{4}$$

Step-by-Step Solution

Answer

Functions f and g are inverses of each other.

Step 1: Find f(g(x))

To find \(f(g(x))\), substitute \(g(x)\) into \(f(x)\). Hence, \(f(g(x)) = f\left(\frac{x}{4}\right) = 4 \cdot \frac{x}{4} = x\)

Step 2: Find g(f(x))

To find \(g(f(x))\), substitute \(f(x)\) into \(g(x)\). Hence, \(g(f(x)) = g(4x) = \frac{4x}{4} = x\)

Step 3: Conclusion

As \(f(g(x)) = x\), and \(g(f(x)) = x\), the functions \(f\) and \(g\) are inverses of each other.