Answer :
To solve for [tex]f(-8)[/tex], we need to identify what is inside the function [tex]f[/tex]. We are given [tex]f(x^3) = 3x^6 + 2x^3[/tex]. This means the input for [tex]f[/tex] is [tex]x^3[/tex]. We want to find what [tex]f(-8)[/tex] is, so we need to express [tex]-8[/tex] in terms of [tex]x^3[/tex].
Step 1: Set [tex]x^3 = -8[/tex]. This implies [tex]x = (-8)^{1/3} = -2[/tex], because [tex](-2)^3 = -8[/tex].
Step 2: Substitute [tex]x = -2[/tex] into the right-hand side of the given function [tex]f(x^3)[/tex]. We need to calculate [tex]f((-2)^3)[/tex] which is the same as evaluating [tex]3(-2)^6 + 2(-2)^3[/tex].
Step 3: Calculate each part of the expression:
- [tex](-2)^6 = 64[/tex], because raising [tex]-2[/tex] to the 6th power yields 64.
- Thus, [tex]3(-2)^6 = 3 \times 64 = 192[/tex].
Step 4: Calculate the second part:
- [tex](-2)^3 = -8[/tex], hence [tex]2(-2)^3 = 2 \times (-8) = -16[/tex].
Step 5: Add the results from Step 3 and Step 4:
- [tex]192 + (-16) = 176[/tex].
Therefore, [tex]f(-8) = 176[/tex], which corresponds to option (1).
The answer is (1) 176.
