Sure, let's break down the given functions and calculate [tex]\( f(g(h(x))) \)[/tex] step by step.
We are given three functions:
1. [tex]\( f(x) = x^4 + 7 \)[/tex]
2. [tex]\( g(x) = x - 9 \)[/tex]
3. [tex]\( h(x) = \sqrt{x} \)[/tex]
We need to find [tex]\( f(g(h(x))) \)[/tex].
### Step 1: Compute [tex]\( h(x) \)[/tex]
The function [tex]\( h(x) = \sqrt{x} \)[/tex].
Let’s consider [tex]\( x = 16 \)[/tex].
Then:
[tex]\[ h(16) = \sqrt{16} = 4 \][/tex]
### Step 2: Compute [tex]\( g(h(x)) \)[/tex]
Using the result from Step 1:
[tex]\[ h(16) = 4 \][/tex]
Now, we find [tex]\( g(h(16)) \)[/tex]:
[tex]\[ g(4) = 4 - 9 = -5 \][/tex]
### Step 3: Compute [tex]\( f(g(h(x))) \)[/tex]
Using the result from Step 2:
[tex]\[ g(h(16)) = -5 \][/tex]
Now, we find [tex]\( f(g(h(16))) \)[/tex]:
[tex]\[ f(-5) = (-5)^4 + 7 = 625 + 7 = 632 \][/tex]
Finally, the value of [tex]\( f(g(h(x))) \)[/tex] when [tex]\( x = 16 \)[/tex] is:
[tex]\[ f(g(h(16))) = 632 \][/tex]
So, the complete process gives us:
- [tex]\( h(16) = 4 \)[/tex]
- [tex]\( g(h(16)) = -5 \)[/tex]
- [tex]\( f(g(h(16))) = 632 \)[/tex]
Hence, the value of [tex]\( f(g(h(x))) \)[/tex] is [tex]\(\boxed{632}\)[/tex].
