Question

let (f(x) = 3x + 11)

(f^{-1}(x) =)

Explanation:

🆕 New Concept Discovered: Inverse Functions
An inverse function undoes the action of the original function, swapping the inputs and outputs.

Step 1: Replace \( f(x) \) with \( y \)

To make the algebra easier to work with, we rewrite the function by replacing \( f(x) \) with \( y \):

$$ y = 3x + 11 $$

Step 2: Swap \( x \) and \( y \)

An inverse function reverses the roles of the input and output. We swap the variables \( x \) and \( y \):

$$ x = 3y + 11 $$

Step 3: Solve for \( y \)

Now, we isolate \( y \) to find the inverse rule.

First, subtract \( 11 \) from both sides:

$$ x - 11 = 3y $$

Next, divide both sides by \( 3 \):

$$ y = \frac{x - 11}{3} $$

Step 4: Write in inverse notation

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

$$ f^{-1}(x) = \frac{x - 11}{3} $$

Answer:

$$ \frac{x - 11}{3} $$