Sure! Let's work through the problem step by step.
### Part a) Find [tex]\( f(2) \)[/tex]
Given the function [tex]\( f(x) = \frac{x+3}{2} \)[/tex]:
To find [tex]\( f(2) \)[/tex], substitute [tex]\( x = 2 \)[/tex] into the function:
[tex]\[
f(2) = \frac{2 + 3}{2}
\][/tex]
Calculating the numerator:
[tex]\[
2 + 3 = 5
\][/tex]
Now, divide the numerator by 2:
[tex]\[
f(2) = \frac{5}{2} = 2.5
\][/tex]
So, [tex]\( f(2) = 2.5 \)[/tex].
### Part b) Find [tex]\( f^{-1}(x) \)[/tex]
To find the inverse function [tex]\( f^{-1}(x) \)[/tex], we start with the equation for the original function and solve for [tex]\( x \)[/tex]:
[tex]\[
y = \frac{x+3}{2}
\][/tex]
First, multiply both sides by 2 to get rid of the fraction:
[tex]\[
2y = x + 3
\][/tex]
Next, isolate [tex]\( x \)[/tex] by subtracting 3 from both sides:
[tex]\[
x = 2y - 3
\][/tex]
So the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = 2x - 3
\][/tex]
### Part c) Find [tex]\( f^{-1}(12) \)[/tex]
Using the inverse function [tex]\( f^{-1}(x) = 2x - 3 \)[/tex], substitute [tex]\( x = 12 \)[/tex]:
[tex]\[
f^{-1}(12) = 2 \cdot 12 - 3
\][/tex]
First, calculate [tex]\( 2 \cdot 12 \)[/tex]:
[tex]\[
2 \cdot 12 = 24
\][/tex]
Then, subtract 3:
[tex]\[
24 - 3 = 21
\][/tex]
So, [tex]\( f^{-1}(12) = 21 \)[/tex].
To summarize:
- [tex]\( f(2) = 2.5 \)[/tex]
- [tex]\( f^{-1}(x) = 2x - 3 \)[/tex]
- [tex]\( f^{-1}(12) = 21 \)[/tex]
