Question

no calculator is allowed on this question.
given ( f(x) = \frac{4 - 2x}{x + 3} ), identify ( f^{-1}(x) ).
select one answer
a ( g(x) = \frac{x + 3}{4 - 2x} )
b ( h(x) = \frac{2x - 4}{x + 3} )
c ( j(x) = \frac{4 + 3x}{x - 2} )
d ( k(x) = \frac{4 - 3x}{x + 2} )

Explanation:

Step1: Let \( y = f(x) \)

We start by setting \( y = \frac{4 - 2x}{x + 3} \).

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

To find the inverse, we swap \( x \) and \( y \), so we get \( x = \frac{4 - 2y}{y + 3} \).

Step3: Solve for \( y \)

First, multiply both sides by \( y + 3 \): \( x(y + 3) = 4 - 2y \)
Expand the left side: \( xy + 3x = 4 - 2y \)
Bring all terms with \( y \) to one side: \( xy + 2y = 4 - 3x \)
Factor out \( y \): \( y(x + 2) = 4 - 3x \)
Then, solve for \( y \): \( y = \frac{4 - 3x}{x + 2} \)
So, \( f^{-1}(x)=\frac{4 - 3x}{x + 2} \), which matches option D.

Answer:

D. \( k(x)=\frac{4 - 3x}{x + 2} \)