To find the inverse function [tex]\( f^{-1}(x) \)[/tex] for the given function [tex]\( f(x) = 3x - 5 \)[/tex], follow these steps carefully:
1. Start with the function:
[tex]\[
f(x) = 3x - 5
\][/tex]
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex] to make manipulation easier:
[tex]\[
y = 3x - 5
\][/tex]
3. Switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. This represents the fundamental operation of finding the inverse, where the roles of the dependent and independent variables are interchanged:
[tex]\[
x = 3y - 5
\][/tex]
4. Solve for [tex]\( y \)[/tex] to find the expression for the inverse function:
[tex]\[
x = 3y - 5
\][/tex]
First, isolate the term involving [tex]\( y \)[/tex]:
[tex]\[
x + 5 = 3y
\][/tex]
Then, solve for [tex]\( y \)[/tex] by dividing both sides by 3:
[tex]\[
y = \frac{x + 5}{3}
\][/tex]
5. Write the inverse function:
[tex]\[
f^{-1}(x) = \frac{x + 5}{3}
\][/tex]
So, the inverse function [tex]\( f^{-1}(x) \)[/tex] of the given function [tex]\( f(x) = 3x - 5 \)[/tex] is:
[tex]\[
f^{-1}(x) = \frac{x}{3} + \frac{5}{3}
\][/tex]
Therefore, the line [tex]\( \square \)[/tex] that represents [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = \frac{x}{3} + \frac{5}{3}
\][/tex]
