Question

cphs: advanced algebra: concepts and connections - block (27.0831030) function inverses 51:2.22 which function is the inverse of ( f(x) = 2x + 3 )? ( f^{-1}(x) = -2x + 3 ) ( f^{-1}(x) = -\frac{1}{2}x - \frac{3}{2} ) ( f^{-1}(x) = 2x + 3 ) ( f^{-1}(x) = \frac{1}{2}x - \frac{3}{2} )

Explanation:

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

We start with the function \( f(x) = 2x + 3 \). Replace \( f(x) \) with \( y \), so we have \( y = 2x + 3 \).

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

To find the inverse, we swap the roles of \( x \) and \( y \). This gives us \( x = 2y + 3 \).

Step3: Solve for \( y \)

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

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

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

Answer:

\( f^{-1}(x)=\frac{1}{2}x-\frac{3}{2} \) (the option with this expression)