Problem 65 - College Algebra

Question

If \(f(x)=3 x\) and \(g(x)=x+5,\) find \((f \circ g)^{-1}(x)\) and \(\left(g^{-1} \circ f^{-1}\right)(x)\)

Step-by-Step Solution

Answer

The inverse of the composed function \((f\circ g)^{-1}(x)\) is \((x/3) - 5\). The function composition of the inverses \((g^{-1}\circ f^{-1})(x)\) is \((x/3) - 5\).

Step 1: Compute \(f \circ g\)

First, compute the function composition \(f\circ g\). To do this, take the function \(f(x)\) and replace \(x\) with \(g(x)\): \[f(g(x)) = 3(g(x)) = 3(x+5).\]

Step 2: Find the Inverse of \(f\circ g\)

To find the inverse of a function \(h(x)\), we replace \(h(x)\) by \(y\), switch x and y in the equation, and solve for \(y\). So, replace \(f(g(x))\) by \(y\), giving \(y = 3(x+5)\). Switch \(x\) and \(y\): \(x = 3(y+5)\). Now, solve for \(y\) to give the inverse function. Solving, we have \(y = (x/3) - 5\). So \((f\circ g)^{-1}(x) = (x/3) - 5\).

Step 3: Find the Inverses of Individual Functions \(f(x)\) and \(g(x)\)

Before we can compute \((g^{-1}\circ f^{-1})\), we need the inverses of \(f(x)\) and \(g(x)\). To find the inverse of \(f(x)=3x\), we proceed as before: Write \(y = 3x\), interchange \(x\) and \(y\) to give \(x = 3y\), and solve for \(y\) to find the inverse function. Here, \(f^{-1}(x) = x/3\). Similarly, for \(g(x) = x + 5\), the inverse function is \(g^{-1}(x) = x - 5\).

Step 4: Compute \(g^{-1} \circ f^{-1}\)

Now let's compute the function composition \(g^{-1}\circ f^{-1}\). We take the function \(g^{-1}(x)\) and replace \(x\) with \(f^{-1}(x)\). This yields: \[g^{-1}(f^{-1}(x)) = (f^{-1}(x) - 5) = ((x/3) - 5).\] Hence, \((g^{-1}\circ f^{-1})(x) = (x/3) - 5\).