To find the inverse function [tex]\( f^{-1}(x) \)[/tex] of the given function [tex]\( f(x) = 3x - 4 \)[/tex], we follow these steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = 3x - 4
\][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
Add 4 to both sides to isolate the term involving [tex]\( x \)[/tex]:
[tex]\[
y + 4 = 3x
\][/tex]
Next, divide both sides by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{y + 4}{3}
\][/tex]
3. Interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to get the inverse function:
[tex]\[
f^{-1}(x) = \frac{x + 4}{3}
\][/tex]
Thus, the inverse of the function [tex]\( f(x) = 3x - 4 \)[/tex] is:
[tex]\[
f^{-1}(x) = \frac{x + 4}{3}
\][/tex]
This corresponds to option D:
[tex]\[
D. \ f^{-1}(x) = \frac{x + 4}{3}
\][/tex]
