Sure, let's go through the process of finding [tex]\((f \circ g)(x)\)[/tex] and then evaluate [tex]\((f \circ g)(2)\)[/tex] step by step.
### Step 1: Define the Functions
We have two functions given:
[tex]\[ f(x) = 3x + 5 \][/tex]
[tex]\[ g(x) = 5 - x \][/tex]
### Step 2: Find [tex]\((f \circ g)(x)\)[/tex]
The composition of [tex]\( f \)[/tex] and [tex]\( g \)[/tex], denoted [tex]\( (f \circ g)(x) \)[/tex], means that we apply [tex]\( g \)[/tex] first and then apply [tex]\( f \)[/tex] to the result of [tex]\( g \)[/tex].
This can be written as:
[tex]\[ (f \circ g)(x) = f(g(x)) \][/tex]
### Step 3: Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]
By substituting [tex]\( g(x) = 5 - x \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f(5 - x) \][/tex]
Now, using the definition of [tex]\( f(x) \)[/tex]:
[tex]\[ f(5 - x) = 3(5 - x) + 5 \][/tex]
### Step 4: Simplify the Expression
Simplify the expression obtained:
[tex]\[ f(5 - x) = 3 \cdot 5 - 3 \cdot x + 5 \][/tex]
[tex]\[ f(5 - x) = 15 - 3x + 5 \][/tex]
[tex]\[ f(5 - x) = 20 - 3x \][/tex]
So, the composition [tex]\( (f \circ g)(x) \)[/tex] is:
[tex]\[ (f \circ g)(x) = 20 - 3x \][/tex]
### Step 5: Evaluate [tex]\((f \circ g)(2)\)[/tex]
To find [tex]\((f \circ g)(2)\)[/tex], we substitute [tex]\(x = 2\)[/tex] into the composition we just found:
[tex]\[ (f \circ g)(2) = 20 - 3 \cdot 2 \][/tex]
[tex]\[ (f \circ g)(2) = 20 - 6 \][/tex]
[tex]\[ (f \circ g)(2) = 14 \][/tex]
### Final Answer
Therefore, the value of [tex]\((f \circ g)(2)\)[/tex] is [tex]\( 14 \)[/tex].
