Problem 88 - College Algebra Essentials

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm working with the linear function \(f(x)=3 x+5\) and \(I\) do not need to find \(f^{-1}\) in order to determine the value of \(\left(f \circ f^{-1}\right)(17)\).

Step-by-Step Solution

Answer

The statement makes sense. The value of \(\left(f \circ f^{-1}\right)(17)\) is 17 without needing to find the explicit form of \(f^{-1}\).

Step 1: Evaluating the statement

Realize that \(\left(f \circ f^{-1}\right)(x)\) represents the function f composed with its inverse \(f^{-1}\). By the definition of the inverse function, the composition \(f(f^{-1}(x))\) should give back the original input x. Therefore, \(\left(f \circ f^{-1}\right)(17)\) should return 17.

Step 2: Explanation of reasoning

The statement makes sense because \(\left(f \circ f^{-1}\right)(x) = x\) for any x in the domain of \(f^{-1}\), without needing to explicitly find the formula for \(f^{-1}\). Thus, \(\left(f \circ f^{-1}\right)(17) = 17\).