Question

question
find $f^{-1}(x)$ if $f(x) = 3x + 5$.

answer attempt 1 out of 99

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$f^{-1}(x) = $
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Explanation:

Step1: Replace \( f(x) \) with \( y \)

We start by writing the function as \( y = 3x + 5 \). This is a standard first step when finding the inverse of a function, as it helps us work with the equation more easily by using \( y \) to represent the output of the function \( f(x) \).

Step2: Swap \( x \) and \( y \)

Next, we interchange the roles of \( x \) and \( y \) to get \( x = 3y + 5 \). This is based on the definition of an inverse function, where if \( y = f(x) \), then \( x = f^{-1}(y) \), so we swap the input and output variables.

Step3: Solve for \( y \)

First, subtract 5 from both sides of the equation: \( x - 5 = 3y \). Then, divide both sides by 3 to isolate \( y \): \( y=\frac{x - 5}{3}\). We can also write this as \( y=\frac{1}{3}x-\frac{5}{3}\).

Step4: Replace \( y \) with \( f^{-1}(x) \)

Since we found \( y \) in terms of \( x \) and \( y = f^{-1}(x) \) by the definition of the inverse function, we substitute back to get \( f^{-1}(x)=\frac{x - 5}{3}\) (or \( \frac{1}{3}x-\frac{5}{3} \)).

Answer:

\( f^{-1}(x)=\frac{x - 5}{3} \) (or equivalent form like \( \frac{1}{3}x-\frac{5}{3} \))