To solve the given problem [tex]\( f\left(\frac{3 x-1}{2}\right)=6 x+1 \)[/tex] and find the value of [tex]\( f(2) \)[/tex], we can follow these steps:
1. Set the expression inside the function equal to 2:
[tex]\[
\frac{3x - 1}{2} = 2
\][/tex]
2. Solve for [tex]\(x\)[/tex]:
- First, multiply both sides of the equation by 2 to clear the fraction:
[tex]\[
3x - 1 = 4
\][/tex]
- Next, add 1 to both sides to isolate the [tex]\(3x\)[/tex] term:
[tex]\[
3x = 5
\][/tex]
- Now, divide both sides by 3 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{5}{3}
\][/tex]
3. Substitute [tex]\(x = \frac{5}{3}\)[/tex] back into the function [tex]\(f\left(\frac{3 x-1}{2}\right)=6 x+1\)[/tex] to find [tex]\( f(2) \)[/tex]:
- The given function can be rewritten with [tex]\(x = \frac{5}{3}\)[/tex]:
[tex]\[
f\left(2\right) = 6 \left(\frac{5}{3}\right) + 1
\][/tex]
- Simplify inside the parenthesis:
[tex]\[
f(2) = 2 \times 5 + 1
\][/tex]
[tex]\[
f(2) = 10 + 1
\][/tex]
[tex]\[
f(2) = 11
\][/tex]
Hence, the value of [tex]\( f(2) \)[/tex] is [tex]\( 11 \)[/tex].
