Problem 95 - 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

\((f \circ g)^{-1}(x) = (x-15)/3\) and \((g^{-1} \circ f^{-1})(x) = (x/3)-5\)

Step 1: Find the Composition of f and g

First calculate the composition of the two functions \(f\) and \(g\) which is denoted as \(f \circ g\). This is done by substituting \(g(x)\) into the function \(f\). Thus, \(f(g(x))=3(g(x))=3(x+5)=3x+15.\)

Step 2: Find the Inverse of the Composition

Next, find \((f \circ g)^{-1}(x)\), the inverse of the composition of \(f\) and \(g\). Firstly, replace \(f(g(x))\) with \(y\), so the equation becomes \(y = 3x + 15\). Then, switch the roles of \(x\) and \(y\) to get \(x = 3y + 15\). Finally, solve the equation for \(y\) which gives \((f \circ g)^{-1}(x) = (x-15)/3\)

Step 3: Find the Inverses of f and g

Now find the inverse of \(f(x)\) and \(g(x)\) separately. For \(f(x)=3x\), replace \(f(x)\) with \(y\), so the equation becomes \(y = 3x\). Swap \(x\) and \(y\) to get \(x = 3y\), and solve for \(y\) to obtain \(f^{-1}(x) = x/3\). Similarly, for \(g(x) = x+5\), the inverse \(g^{-1}(x) = x - 5 \).

Step 4: Calculate the Composition of the Inverses

Finally, calculate the composition of the inverses as \((g^{-1} \circ f^{-1})(x)\). Insert the inverse of \(f(x)\) into \(g^{-1}(x)\), to obtain \(g^{-1}(f^{-1}(x)) = (x/3)-5\)