Certainly! Let's find [tex]\( f(5) \)[/tex] given the function [tex]\( f(x + 2) = 6x^3 + 5x - 8 \)[/tex].
1. Identify the relation: We know that [tex]\( f(x+2) \)[/tex] represents the function in terms of [tex]\( x \)[/tex].
2. Set up the equation: To find [tex]\( f(5) \)[/tex], we need to determine the [tex]\( x \)[/tex] value that makes [tex]\( x + 2 = 5 \)[/tex].
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
x + 2 = 5
\][/tex]
Subtract 2 from each side:
[tex]\[
x = 3
\][/tex]
4. Substitute [tex]\( x \)[/tex] back into the function: Now we substitute [tex]\( x = 3 \)[/tex] into the function [tex]\( f(x+2) \)[/tex].
5. Evaluate the function:
[tex]\[
f(5) = 6 \cdot 3^3 + 5 \cdot 3 - 8
\][/tex]
6. Calculate the individual terms:
- [tex]\( 3^3 = 27 \)[/tex]
- [tex]\( 6 \cdot 27 = 162 \)[/tex]
- [tex]\( 5 \cdot 3 = 15 \)[/tex]
7. Combine the terms:
[tex]\[
162 + 15 - 8
\][/tex]
8. Perform the addition and subtraction:
- [tex]\( 162 + 15 = 177 \)[/tex]
- [tex]\( 177 - 8 = 169 \)[/tex]
Therefore, the value of [tex]\( f(5) \)[/tex] is [tex]\( \boxed{169} \)[/tex].
