Given that [tex]f(x)=7 x+7[/tex] and [tex]g(x)=4-x^2[/tex], calculate
(a) [tex]f(g(0))=[/tex] $\square$
(b) [tex]g(f(0))=[/tex] $\square$
Answer (1)
Calculate g ( 0 ) : g ( 0 ) = 4 − 0 2 = 4 .
Calculate f ( g ( 0 )) = f ( 4 ) : f ( 4 ) = 7 × 4 + 7 = 35 .
Calculate f ( 0 ) : f ( 0 ) = 7 × 0 + 7 = 7 .
Calculate g ( f ( 0 )) = g ( 7 ) : g ( 7 ) = 4 − 7 2 = − 45 .
The final answers are 35 and − 45 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 7 x + 7 and g ( x ) = 4 − x 2 . We need to find the values of f ( g ( 0 )) and g ( f ( 0 )) . This involves evaluating the functions at specific points and then composing them.
Calculating g(0) First, we need to find g ( 0 ) . We substitute x = 0 into the expression for g ( x ) : g ( 0 ) = 4 − ( 0 ) 2 = 4 − 0 = 4
Calculating f(g(0)) Now we can find f ( g ( 0 )) , which is f ( 4 ) . We substitute x = 4 into the expression for f ( x ) : f ( 4 ) = 7 ( 4 ) + 7 = 28 + 7 = 35
Calculating f(0) Next, we need to find f ( 0 ) . We substitute x = 0 into the expression for f ( x ) : f ( 0 ) = 7 ( 0 ) + 7 = 0 + 7 = 7
Calculating g(f(0)) Finally, we can find g ( f ( 0 )) , which is g ( 7 ) . We substitute x = 7 into the expression for g ( x ) : g ( 7 ) = 4 − ( 7 ) 2 = 4 − 49 = − 45
Final Answer Therefore, f ( g ( 0 )) = 35 and g ( f ( 0 )) = − 45 .
Examples
Function composition 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 f ( x ) = 0.9 x represents the discount and g ( x ) = 1.05 x represents the sales tax, then g ( f ( x )) represents the final price of an item after both the discount and the sales tax are applied. Understanding function composition allows us to model and analyze such scenarios effectively.
