To find the inverse of the function [tex]\( f(x) = \frac{3x - 5}{7} \)[/tex], let's follow a step-by-step process:
1. Express the function in terms of [tex]\( y \)[/tex]:
[tex]\[
y = \frac{3x - 5}{7}
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to begin solving for the inverse function:
[tex]\[
x = \frac{3y - 5}{7}
\][/tex]
3. Isolate [tex]\( y \)[/tex]. First, we multiply both sides of the equation by 7 to get rid of the denominator:
[tex]\[
7x = 3y - 5
\][/tex]
4. Solve for [tex]\( y \)[/tex] by adding 5 to both sides:
[tex]\[
7x + 5 = 3y
\][/tex]
5. Divide both sides by 3 to isolate [tex]\( y \)[/tex]:
[tex]\[
y = \frac{7x + 5}{3}
\][/tex]
Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = \frac{7x + 5}{3}
\][/tex]
So, the correct answer is:
[tex]\[
f^{-1}(x) = \frac{7x + 5}{3}
\][/tex]
