Question

identify the inverse to the function $f(x)=\frac{3x}{4}+\frac{1}{2}$. (5 points)○ $f^{-1}(x)=\frac{4x}{3}-\frac{1}{2}$○ $f^{-1}(x)=\frac{4x-2}{3}$○ $f^{-1}(x)=\frac{3x}{4}-\frac{1}{2}$

Explanation:

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

$y = \frac{3x}{4} + \frac{1}{2}$

Step2: Swap $x$ and $y$

$x = \frac{3y}{4} + \frac{1}{2}$

Step3: Isolate the term with $y$

$x - \frac{1}{2} = \frac{3y}{4}$

Step4: Solve for $y$

First, rewrite $x - \frac{1}{2}$ as $\frac{2x - 1}{2}$. Then:
$\frac{2x - 1}{2} = \frac{3y}{4}$
Multiply both sides by 4: $2(2x - 1) = 3y$
Expand: $4x - 2 = 3y$
Divide by 3: $y = \frac{4x - 2}{3}$

Answer:

$f^{-1}(x)=\frac{4x-2}{3}$