Certainly! Let's break this down step-by-step to find [tex]\( g(h(t)) \)[/tex].
### Step 1: Understand the Functions
Given the functions:
[tex]\[ g(t) = 2t + 3 \][/tex]
[tex]\[ h(t) = 3t - 4 \][/tex]
### Step 2: Substitute [tex]\( h(t) \)[/tex] into [tex]\( g \)[/tex]
We need to find [tex]\( g(h(t)) \)[/tex]. This means we will substitute the function [tex]\( h(t) \)[/tex] into the function [tex]\( g(t) \)[/tex].
### Step 3: Substitute [tex]\( h(t) \)[/tex] into [tex]\( g(t) \)[/tex]
First, we rewrite [tex]\( g(t) \)[/tex] so that it accepts [tex]\( h(t) \)[/tex]:
[tex]\[ g(t) = 2t + 3 \][/tex]
When substituting [tex]\( h(t) \)[/tex], we replace [tex]\( t \)[/tex] in [tex]\( g(t) \)[/tex] with [tex]\( h(t) \)[/tex]:
[tex]\[ g(h(t)) = g(3t - 4) \][/tex]
### Step 4: Evaluate [tex]\( g(3t - 4) \)[/tex]
Now, substitute [tex]\( 3t - 4 \)[/tex] wherever there is a [tex]\( t \)[/tex] in the function [tex]\( g(t) \)[/tex]:
[tex]\[ g(3t - 4) = 2(3t - 4) + 3 \][/tex]
### Step 5: Simplify the Expression
Next, we simplify the expression step-by-step:
[tex]\[ 2(3t - 4) + 3 \][/tex]
[tex]\[ = 2 \cdot 3t - 2 \cdot 4 + 3 \][/tex]
[tex]\[ = 6t - 8 + 3 \][/tex]
[tex]\[ = 6t - 5 \][/tex]
So, [tex]\( g(h(t)) = 6t - 5 \)[/tex].
### Final Answer:
To find the specific value of [tex]\( g(h(t)) \)[/tex] for a particular [tex]\( t \)[/tex], you would substitute the value of [tex]\( t \)[/tex] into [tex]\( 6t - 5 \)[/tex]. For example, if [tex]\( t = 0 \)[/tex]:
[tex]\[ g(h(0)) = 6(0) - 5 = -5 \][/tex]
Therefore, the composite function [tex]\( g(h(t)) \)[/tex] simplifies to [tex]\( 6t - 5 \)[/tex]. For [tex]\( t = 0 \)[/tex], [tex]\( g(h(0)) = -5 \)[/tex].
