To find the inverse of the function [tex]\( f(x) = \frac{3x - 2}{6} \)[/tex], we will follow these steps:
1. Rewrite the function using [tex]\(y\)[/tex].
[tex]\[
y = \frac{3x - 2}{6}
\][/tex]
2. Swap the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex] and solve for [tex]\(y\)[/tex].
[tex]\[
x = \frac{3y - 2}{6}
\][/tex]
3. Solve for [tex]\(y\)[/tex].
[tex]\[
x = \frac{3y - 2}{6}
\][/tex]
Multiply both sides by 6 to get rid of the fraction:
[tex]\[
6x = 3y - 2
\][/tex]
Add 2 to both sides:
[tex]\[
6x + 2 = 3y
\][/tex]
Divide both sides by 3:
[tex]\[
y = \frac{6x + 2}{3}
\][/tex]
So, the inverse function of [tex]\( f(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = \frac{6x + 2}{3}
\][/tex]
4. Verify which option matches this result.
- Option A: [tex]\( f^{-1}(x) = \frac{6x - 2}{3} \)[/tex]
- Option B: [tex]\( f^{-1}(x) = \frac{6x + 2}{3} \)[/tex]
- Option C: [tex]\( f^{-1}(x) = \frac{2x + 3}{6} \)[/tex]
- Option D: [tex]\( f^{-1}(x) = \frac{2x - 3}{6} \)[/tex]
The correct answer is option B:
[tex]\[
f^{-1}(x) = \frac{6x + 2}{3}
\][/tex]
Thus, the inverse of [tex]\(f(x) = \frac{3x - 2}{6} \)[/tex] is [tex]\( f^{-1}(x) = \frac{6x + 2}{3} \)[/tex], which corresponds to option B.
