b) If [tex]f(x)=x^2+2 x+1, g(x)=-2 x+3[/tex] and [tex]h(x)=4 x[/tex], find:
i) [tex](f \circ g)(-2)[/tex]
ii) [tex](g \circ f)(-2)[/tex]
Answer (1)
First, calculate g ( − 2 ) = − 2 ( − 2 ) + 3 = 7 .
Then, calculate f ( g ( − 2 )) = f ( 7 ) = 7 2 + 2 ( 7 ) + 1 = 64 .
Next, calculate f ( − 2 ) = ( − 2 ) 2 + 2 ( − 2 ) + 1 = 1 .
Finally, calculate g ( f ( − 2 )) = g ( 1 ) = − 2 ( 1 ) + 3 = 1 . Thus, ( f ∘ g ) ( − 2 ) = 64 and ( g ∘ f ) ( − 2 ) = 1 , so the answers are 64 , 1 .
Explanation
Understanding the Problem We are given three functions: f ( x ) = x 2 + 2 x + 1 , g ( x ) = − 2 x + 3 , and h ( x ) = 4 x . We need to find ( f ∘ g ) ( − 2 ) and ( g ∘ f ) ( − 2 ) . The composition of functions f ∘ g is defined as f ( g ( x )) .
Calculating g(-2) To find ( f ∘ g ) ( − 2 ) , we first need to evaluate g ( − 2 ) . Substituting − 2 into g ( x ) , we get: g ( − 2 ) = − 2 ( − 2 ) + 3 = 4 + 3 = 7
Calculating f(g(-2)) Now, we substitute the result g ( − 2 ) = 7 into f ( x ) to find f ( g ( − 2 )) :
f ( g ( − 2 )) = f ( 7 ) = ( 7 ) 2 + 2 ( 7 ) + 1 = 49 + 14 + 1 = 64
Calculating f(-2) Next, we need to find ( g ∘ f ) ( − 2 ) . First, we evaluate f ( − 2 ) . Substituting − 2 into f ( x ) , we get: f ( − 2 ) = ( − 2 ) 2 + 2 ( − 2 ) + 1 = 4 − 4 + 1 = 1
Calculating g(f(-2)) Now, we substitute the result f ( − 2 ) = 1 into g ( x ) to find g ( f ( − 2 )) :
g ( f ( − 2 )) = g ( 1 ) = − 2 ( 1 ) + 3 = − 2 + 3 = 1
Final Answer Therefore, ( f ∘ g ) ( − 2 ) = 64 and ( g ∘ f ) ( − 2 ) = 1 .
Examples
Function composition is a fundamental concept in mathematics and has many real-world applications. For example, consider a store that applies a discount and then adds sales tax. If d ( x ) represents the discounted price of an item originally priced at x , and t ( x ) represents the price after sales tax is added, then t ( d ( x )) represents the final price of the item. Understanding function composition allows businesses to model and optimize pricing strategies, calculate costs, and analyze the impact of various discounts and taxes on their revenue.
