To find the inverse of the function [tex]\( y = 2x + \frac{5}{2} \)[/tex], we need to follow several steps. Let’s solve this step-by-step:
1. Express the function as [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y = 2x + \frac{5}{2}
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. This step is crucial because finding the inverse function entails reversing the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = 2y + \frac{5}{2}
\][/tex]
3. Solve for [tex]\( y \)[/tex]. To isolate [tex]\( y \)[/tex], we proceed as follows:
[tex]\[
x = 2y + \frac{5}{2}
\][/tex]
Subtract [tex]\(\frac{5}{2}\)[/tex] from both sides:
[tex]\[
x - \frac{5}{2} = 2y
\][/tex]
4. Divide both sides by 2 to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{x - \frac{5}{2}}{2}
\][/tex]
Simplify the expression inside the fraction:
[tex]\[
y = \frac{1}{2} (x - \frac{5}{2})
\][/tex]
Distribute the [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
y = \frac{1}{2} x - \frac{1}{2} \cdot \frac{5}{2}
\][/tex]
[tex]\[
y = \frac{1}{2} x - \frac{5}{4}
\][/tex]
Therefore, the inverse function is:
[tex]\[
y = \frac{1}{2} x - \frac{5}{4}
\][/tex]
Among the given options, the correct one is:
C. [tex]\( y = \frac{1}{2} x - \frac{5}{4} \)[/tex]
