To solve the problem, follow these steps:
Step 1: Understand the given functional equation.
We are given:
[tex]\[ f(3x - 5) = 15x + 8 \][/tex]
Step 2: Introduce a substitution to simplify the expression.
Let [tex]\( u = 3x - 5 \)[/tex].
Step 3: Rewrite the given equation using [tex]\( u \)[/tex].
Since [tex]\( u = 3x - 5 \)[/tex], we can solve for [tex]\( x \)[/tex] in terms of [tex]\( u \)[/tex]:
[tex]\[ x = \frac{u + 5}{3} \][/tex]
Step 4: Express [tex]\( f(u) \)[/tex] using this substitution.
Replace [tex]\( x \)[/tex] in [tex]\( 15x + 8 \)[/tex]:
[tex]\[ f(u) = 15 \left(\frac{u + 5}{3}\right) + 8 \][/tex]
Step 5: Simplify the expression for [tex]\( f(u) \)[/tex].
First, distribute the 15:
[tex]\[ f(u) = 15 \cdot \frac{u + 5}{3} + 8 \][/tex]
[tex]\[ f(u) = 5(u + 5) + 8 \][/tex]
Next, distribute the 5:
[tex]\[ f(u) = 5u + 25 + 8 \][/tex]
Combine like terms:
[tex]\[ f(u) = 5u + 33 \][/tex]
Step 6: Replace [tex]\( u \)[/tex] back with [tex]\( x \)[/tex].
Since [tex]\( u = x \)[/tex] in the function [tex]\( f(u) \)[/tex], we have:
[tex]\[ f(x) = 5x + 33 \][/tex]
Step 7: Calculate [tex]\( f(2) \)[/tex].
Substitute [tex]\( x = 2 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(2) = 5(2) + 33 \][/tex]
[tex]\[ f(2) = 10 + 33 \][/tex]
[tex]\[ f(2) = 43 \][/tex]
Final Results:
- The function [tex]\( f(x) \)[/tex] is:
[tex]\[ f(x) = 5x + 33 \][/tex]
- The value of [tex]\( f(2) \)[/tex] is:
[tex]\[ f(2) = 43 \][/tex]
