Sure, let's solve these functions step-by-step as requested.
### Problem 9: [tex]\( h[f(-2)] \)[/tex]
First, we need to evaluate [tex]\( f(-2) \)[/tex]:
[tex]\[ f(x) = x^2 \][/tex]
[tex]\[ f(-2) = (-2)^2 = 4 \][/tex]
Next, we substitute this result into function [tex]\( h \)[/tex]:
[tex]\[ h(x) = x^2 - 1 \][/tex]
[tex]\[ h[f(-2)] = h(4) = 4^2 - 1 = 16 - 1 = 15 \][/tex]
So, [tex]\( h[f(-2)] = 15 \)[/tex].
### Problem 10: [tex]\( f[g(x-1)] \)[/tex]
First, we need to evaluate [tex]\( g(x-1) \)[/tex]:
[tex]\[ g(x) = 2x + 5 \][/tex]
[tex]\[ g(x-1) = 2(x-1) + 5 = 2x - 2 + 5 = 2x + 3 \][/tex]
Next, we substitute this expression into function [tex]\( f \)[/tex]:
[tex]\[ f(x) = x^2 \][/tex]
[tex]\[ f[g(x-1)] = f(2x + 3) = (2x + 3)^2 \][/tex]
We then expand the square:
[tex]\[ (2x + 3)^2 = (2x + 3)(2x + 3) = 4x^2 + 12x + 9 \][/tex]
So, [tex]\( f[g(x-1)] = 4x^2 + 12x + 9 \)[/tex].
### Problem 11: [tex]\( g\left[h\left(x^3\right)\right] \)[/tex]
First, we need to evaluate [tex]\( h(x^3) \)[/tex]:
[tex]\[ h(x) = x^2 - 1 \][/tex]
[tex]\[ h(x^3) = (x^3)^2 - 1 = x^6 - 1 \][/tex]
Next, we substitute this expression into function [tex]\( g \)[/tex]:
[tex]\[ g(x) = 2x + 5 \][/tex]
[tex]\[ g\left[h\left(x^3\right)\right] = g(x^6 - 1) = 2(x^6 - 1) + 5 \][/tex]
We'll simplify the expression:
[tex]\[ 2(x^6 - 1) + 5 = 2x^6 - 2 + 5 = 2x^6 + 3 \][/tex]
So, [tex]\( g\left[h\left(x^3\right)\right] = 2x^6 + 3 \)[/tex].
To summarize:
1. [tex]\( h[f(-2)] = 15 \)[/tex]
2. [tex]\( f[g(x-1)] = 4x^2 + 12x + 9 \)[/tex]
3. [tex]\( g\left[h\left(x^3\right)\right] = 2x^6 + 3 \)[/tex]
