Question

complete the following sentence.

if \\(f\\) is a one-to-one function and \\(f(8) = 1\\), then \\(f^{-1}(1) =\\) .

Explanation:

Identify the given information

We are given that \(f\) is a one-to-one function and \(f(8) = 1\). We need to find the value of \(f^{-1}(1)\).

Apply the definition of an inverse function

Using the Definition of a Function concept, we know that a function maps an input to a unique output. For a one-to-one function, its inverse function \(f^{-1}\) reverses this mapping. Specifically, the definition of an inverse function states:

$$ f(x) = y \iff f^{-1}(y) = x $$

Evaluate the inverse function

Using the relationship above, since we are given:

$$ f(8) = 1 $$

it directly follows that:

$$ f^{-1}(1) = 8 $$

Answer:

Complete the following sentence.

If f is a one-to-one function and f(8) = 1, then \(f^{-1}(1) =\) <blank>8</blank>.