To find the composition [tex]\((g \circ f \circ h)(x)\)[/tex], we need to apply each function successively. Let's break it down step-by-step.
### Step 1: Define the functions
We have three functions given:
- [tex]\( f(x) = 3x - 1 \)[/tex]
- [tex]\( g(x) = x^3 \)[/tex]
- [tex]\( h(x) = x \)[/tex] (Assuming [tex]\( h(x) = x \)[/tex] as it is not provided explicitly)
### Step 2: Compose [tex]\( f \)[/tex] and [tex]\( h \)[/tex]
First, we need to find [tex]\( f(h(x)) \)[/tex]:
[tex]\[ f(h(x)) = f(x) \][/tex]
[tex]\[ f(x) = 3x - 1 \][/tex]
So,
[tex]\[ f(h(x)) = 3x - 1 \][/tex]
### Step 3: Compose [tex]\( g \)[/tex] and [tex]\( f(h(x)) \)[/tex]
Next, we need to find [tex]\( g(f(h(x))) \)[/tex]:
[tex]\[ g(y) = y^3 \][/tex]
where [tex]\( y = f(h(x)) \)[/tex].
From Step 2, we know:
[tex]\[ f(h(x)) = 3x - 1 \][/tex]
Thus,
[tex]\[ g(f(h(x))) = g(3x - 1) \][/tex]
[tex]\[ g(3x - 1) = (3x - 1)^3 \][/tex]
### Final Result
Combining all the steps, we get:
[tex]\[ (g \circ f \circ h)(x) = (3x - 1)^3 \][/tex]
Hence, the composed function [tex]\((g \circ f \circ h)(x) \)[/tex] is:
[tex]\[ (g \circ f \circ h)(x) = (3x - 1)^3 \][/tex]
