Question

find $f^{-1}(x)$, the inverse of the function $f(x) = 4x + 6$. drag numbers into the blanks to complete the expression for $f^{-1}(x)$. $f^{-1}(x) = \underline{quadquad}x + \underline{quadquad}$ the numbers to choose from: $-\frac{3}{2}$, $-\frac{3}{4}$, $-\frac{2}{3}$, $-\frac{1}{4}$, $\frac{1}{4}$, $\frac{2}{3}$, $\frac{3}{4}$, $\frac{3}{2}$

Explanation:

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

We start with the function \( f(x) = 4x + 6 \). Replace \( f(x) \) with \( y \), so we have \( y = 4x + 6 \).

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

To find the inverse, we swap the roles of \( x \) and \( y \). This gives us \( x = 4y + 6 \).

Step3: Solve for \( y \)

First, subtract 6 from both sides of the equation: \( x - 6 = 4y \). Then, divide both sides by 4 to isolate \( y \): \( y=\frac{x - 6}{4}=\frac{1}{4}x-\frac{6}{4}=\frac{1}{4}x-\frac{3}{2}\).

Answer:

\( f^{-1}(x)=\frac{1}{4}x-\frac{3}{2} \) (So the first blank is \(\frac{1}{4}\) and the second blank is \(-\frac{3}{2}\))