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 (1)
We are given f ( x ) = 3 x + 2 and g ( x ) = x 2 + 1 . We want to find ( f ∘ g ) ( x ) = f ( g ( x )) .
First, we find g ( x ) = x 2 + 1 .
Then, we substitute g ( x ) into f ( x ) to get f ( g ( x )) = 3 ( x 2 + 1 ) + 2 .
Finally, we simplify the expression to get 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 that is equivalent to ( f ∘ g ) ( x ) , which means f ( g ( x )) .
Finding g(x) First, we need to find g ( x ) , which is given as 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 ) with g ( x ) , 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 .
Finding the Equivalent Expression Finally, we compare our simplified expression 3 x 2 + 5 with the given options to find the equivalent expression. The equivalent expression is 3 ( x 2 + 1 ) + 2 .
Examples
Understanding function composition is crucial in many areas, such as computer graphics, where transformations are applied sequentially. For example, rotating an image and then scaling it can be represented as a composition of two functions. Similarly, in calculus, the chain rule is used to differentiate composite functions, allowing us to analyze complex systems by breaking them down into simpler components. This concept helps in modeling real-world processes where multiple operations are performed in sequence.
