Which function is the inverse of [tex][tex]$y=2x+\frac{5}{2}$[/tex][/tex]?

A. [tex]y=x-\frac{5}{4}[/tex]
B. [tex]y=-2x-\frac{5}{2}[/tex]
C. [tex]y=\frac{1}{2}x-\frac{5}{4}[/tex]
D. [tex]y=\frac{1}{2}x+\frac{2}{5}[/tex]



Answer :

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]