Problem 9 - College Algebra

Question

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

Step-by-Step Solution

Answer

The function compositions \(f(g(x))\) and \(g(f(x))\) are both \(x\), hence the given functions \(f(x)=-x\) and \(g(x)=-x\) are inverses of each other.

Step 1: Finding \(f(g(x))\)

To find \(f(g(x))\), \(g(x)\) should be substituted into \(f(x)\). Here, \(g(x)=-x\) is substituted into \(f(x)=-x\), yielding \(f(g(x))= -(-x) = x.\)

Step 2: Finding \(g(f(x))\)

Similarly, to find \(g(f(x))\), \(f(x)\) should be substituted into \(g(x)\). Here, \(f(x)=-x\) is substituted into \(g(x)=-x\), giving \(g(f(x)) = -(-x) = x.\)

Step 3: Determining if \(f\) and \(g\) are inverses

Two functions are inverses of each other if and only if \(f(g(x)) = g(f(x)) = x\). From the previous steps, both \(f(g(x))\) and \(g(f(x))\) have been found to equal \(x\), so \(f\) and \(g\) are inverses.