Certainly! Let's go through the steps to find the inverse function of [tex]\( f(x) = 5x - 3 \)[/tex].
### Step 1: Substitute [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]
We start by rewriting the function [tex]\( f(x) = 5x - 3 \)[/tex] with [tex]\( y \)[/tex] instead of [tex]\( f(x) \)[/tex]:
[tex]\[ y = 5x - 3 \][/tex]
### Step 2: Solve for [tex]\( x \)[/tex]
To find the inverse function, we need to solve this equation for [tex]\( x \)[/tex]:
1. Add 3 to both sides of the equation to isolate the term containing [tex]\( x \)[/tex]:
[tex]\[ y + 3 = 5x \][/tex]
2. Divide both sides by 5 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{y + 3}{5} \][/tex]
### Step 3: Express the Inverse Function
Now that we have solved for [tex]\( x \)[/tex], we write the inverse function [tex]\( f^{-1}(y) \)[/tex]. In mathematics, we usually denote the inverse function with [tex]\( x \)[/tex] again for the variable, following the convention [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[ f^{-1}(x) = \frac{x + 3}{5} \][/tex]
### Conclusion
The correct expression representing the inverse function of [tex]\( f(x) = 5x - 3 \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{x + 3}{5} \][/tex]
Looking at the given choices:
A. [tex]\(\frac{(x+3)}{(x+2)}\)[/tex]
B. [tex]\(\frac{(x+3)}{(5)}\)[/tex]
C. [tex]\(\frac{(5)}{(x+3)}\)[/tex]
D. [tex]\(\frac{(x-3)}{(2)}\)[/tex]
The correct answer is:
B. [tex]\(\frac{(x+3)}{(5)}\)[/tex]
So, the answer is:
[tex]\[ \boxed{\frac{(x+3)}{(5)}} \][/tex]
