To solve the problem, follow these steps:
1. Define the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
- [tex]\( f(x) = 3x - 1 \)[/tex]
- [tex]\( g(x) = 2x - 3 \)[/tex]
2. Calculate [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = 3 \cdot 2 - 1 = 6 - 1 = 5
\][/tex]
3. Evaluate [tex]\( g(x) - f(2) \)[/tex] for each given value of [tex]\( x \)[/tex]:
For [tex]\( x = \frac{3}{2} \)[/tex]:
[tex]\[
g\left(\frac{3}{2}\right) = 2 \cdot \frac{3}{2} - 3 = 3 - 3 = 0
\][/tex]
[tex]\[
g\left(\frac{3}{2}\right) - f(2) = 0 - 5 = -5
\][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[
g(2) = 2 \cdot 2 - 3 = 4 - 3 = 1
\][/tex]
[tex]\[
g(2) - f(2) = 1 - 5 = -4
\][/tex]
For [tex]\( x = \frac{5}{2} \)[/tex]:
[tex]\[
g\left(\frac{5}{2}\right) = 2 \cdot \frac{5}{2} - 3 = 5 - 3 = 2
\][/tex]
[tex]\[
g\left(\frac{5}{2}\right) - f(2) = 2 - 5 = -3
\][/tex]
For [tex]\( x = 4 \)[/tex]:
[tex]\[
g(4) = 2 \cdot 4 - 3 = 8 - 3 = 5
\][/tex]
[tex]\[
g(4) - f(2) = 5 - 5 = 0
\][/tex]
4. Combine the results:
- For [tex]\( x = \frac{3}{2} \)[/tex], [tex]\( g\left(\frac{3}{2}\right) - f(2) = -5 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( g(2) - f(2) = -4 \)[/tex]
- For [tex]\( x = \frac{5}{2} \)[/tex], [tex]\( g\left(\frac{5}{2}\right) - f(2) = -3 \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( g(4) - f(2) = 0 \)[/tex]
So, the result for each value of [tex]\( x \)[/tex] are:
[tex]\[
x = \frac{3}{2} \rightarrow -5
\][/tex]
[tex]\[
x = 2 \rightarrow -4
\][/tex]
[tex]\[
x = \frac{5}{2} \rightarrow -3
\][/tex]
[tex]\[
x = 4 \rightarrow 0
\][/tex]
