Question

understanding inverse functions
what is the inverse of the function ( f(x) = 2x + 6 )?
solve ( y )
( y \to f(x) )

Explanation:

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

We start with the function \( f(x) = 2x + 6 \). Replace \( f(x) \) with \( y \), so we get \( y = 2x + 6 \).

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

To find the inverse, we swap the roles of \( x \) and \( y \). This gives us \( x = 2y + 6 \).

Step3: Solve for \( y \)

First, subtract 6 from both sides of the equation: \( x - 6 = 2y \). Then, divide both sides by 2: \( y = \frac{x - 6}{2} \), which can be simplified to \( y=\frac{1}{2}x - 3 \).

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

Now, we replace \( y \) with the inverse function notation \( f^{-1}(x) \). So, \( f^{-1}(x)=\frac{1}{2}x - 3 \).

Answer:

The inverse of the function \( f(x) = 2x + 6 \) is \( f^{-1}(x)=\frac{1}{2}x - 3 \)