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 g ( x ) : g ( x ) = x 2 + 1 .
Substitute g ( x ) into f ( x ) : f ( g ( x )) = 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 want to find the expression equivalent to ( f c i rc g ) ( x ) , which means we need to find f ( g ( x )) . In other words, we need to substitute g ( x ) into f ( x ) .
Identifying g(x) First, we identify g ( x ) = x 2 + 1 .
Substituting g(x) into f(x) Next, we substitute g ( x ) into f ( x ) . This means we replace x in f ( x ) = 3 x + 2 with ( x 2 + 1 ) . So we have
f ( g ( x )) = f ( x 2 + 1 ) = 3 ( x 2 + 1 ) + 2
Simplifying the Expression Now, we simplify the expression:
3 ( x 2 + 1 ) + 2 = 3 x 2 + 3 + 2 = 3 x 2 + 5
Checking the Options Therefore, ( f c i rc g ) ( x ) = 3 x 2 + 5 . Now we need to check which of the given options is equivalent to 3 x 2 + 5 .
( 3 x + 2 ) ( x 2 + 1 ) = 3 x 3 + 3 x + 2 x 2 + 2 (Not equivalent) 3 x 2 + 1 + 2 = 3 x 2 + 3 (Not equivalent) ( 3 x + 2 ) 2 + 1 = 9 x 2 + 12 x + 4 + 1 = 9 x 2 + 12 x + 5 (Not equivalent) 3 ( x 2 + 1 ) + 2 = 3 x 2 + 3 + 2 = 3 x 2 + 5 (Equivalent)
Final Answer Thus, the expression equivalent to ( f c i rc g ) ( x ) is 3 ( x 2 + 1 ) + 2 .
Examples
Composition of functions is a fundamental concept in mathematics and has many real-world applications. For example, consider a store that offers a discount of 10% on all items and then applies a sales tax of 5%. If d ( x ) = 0.9 x represents the price after the discount and t ( x ) = 1.05 x represents the price after tax, then the final price can be represented by the composition t ( d ( x )) , which means applying the discount first and then the tax. Understanding function composition helps in modeling such scenarios and predicting outcomes.
To find ( f ∘ g ) ( x ) , we substituted g ( x ) = x 2 + 1 into f ( x ) = 3 x + 2 to get f ( g ( x )) = 3 ( x 2 + 1 ) + 2 , which simplifies to 3 x 2 + 5 . The equivalent expression from the options provided is therefore option D: 3(x^2 + 1) + 2 .
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