If [tex]f(x)=3 x+2[/tex] and [tex]g(x)=x^2+1[/tex], which expression is equivalent to [tex](f \circ g)(x)[/tex]?
A. [tex](3 x+2)(x^2+1)[/tex]
B. [tex]3 x^2+1+2[/tex]
C. [tex](3 x+2)^2+1[/tex]
D. [tex]3(x^2+1)+2[/tex]
Answer (2)
Find ( f ∘ g ) ( x ) by substituting g ( x ) into f ( x ) .
Substitute g ( x ) = x 2 + 1 into f ( x ) = 3 x + 2 , resulting in f ( x 2 + 1 ) = 3 ( x 2 + 1 ) + 2 .
Simplify the expression: 3 ( x 2 + 1 ) + 2 = 3 x 2 + 3 + 2 = 3 x 2 + 5 .
The equivalent expression is 3 ( x 2 + 1 ) + 2 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 3 x + 2 and g ( x ) = x 2 + 1 . We need to find the expression that is equivalent to the composition of these two functions, ( f ∘ g ) ( x ) . The composition ( f ∘ g ) ( x ) means we need to substitute the function g ( x ) into the function f ( x ) .
Substituting g(x) into f(x) To find ( f ∘ g ) ( x ) , we substitute g ( x ) into f ( x ) . So, we have f ( g ( x )) = f ( x 2 + 1 ) . This means we replace the x in f ( x ) with the expression x 2 + 1 .
Performing the Substitution Now we substitute x 2 + 1 into f ( x ) = 3 x + 2 . We get f ( x 2 + 1 ) = 3 ( x 2 + 1 ) + 2 .
Simplifying the Expression Next, we simplify the expression 3 ( x 2 + 1 ) + 2 .
First, distribute the 3 : 3 ( x 2 + 1 ) = 3 x 2 + 3 .
Then, add the 2 : 3 x 2 + 3 + 2 = 3 x 2 + 5 .
Final Answer Therefore, ( f ∘ g ) ( x ) = 3 x 2 + 5 . Comparing this to the given options, we see that 3 ( x 2 + 1 ) + 2 is the correct expression.
Examples
Function composition is a fundamental concept in mathematics and has many real-world applications. For example, consider a store that marks up the price of its products by 20%, and then applies a 5% sales tax. If m ( x ) = 1.20 x represents the markup function and t ( x ) = 1.05 x represents the sales tax function, then the final price of a product can be represented by the composition ( t ∘ m ) ( x ) = t ( m ( x )) = 1.05 ( 1.20 x ) = 1.26 x . This means the final price is 126% of the original price.
The expression equivalent to ( f ∘ g ) ( x ) is 3 ( x 2 + 1 ) + 2 , which simplifies to 3 x 2 + 5 . This corresponds to option D in the provided choices. Thus, D is the correct answer.
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