Given that $f(x)=3 x-9$ and $g(x)=1-x^2$, calculate
(a) $f(g(0))=$ $\square$
(b) $g(f(0))=$ $\square$
Answer (1)
Calculate g ( 0 ) = 1 − 0 2 = 1 .
Calculate f ( g ( 0 )) = f ( 1 ) = 3 ( 1 ) − 9 = − 6 .
Calculate f ( 0 ) = 3 ( 0 ) − 9 = − 9 .
Calculate g ( f ( 0 )) = g ( − 9 ) = 1 − ( − 9 ) 2 = − 80 .
The final answers are − 6 and − 80 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 3 x − 9 and g ( x ) = 1 − x 2 . Our goal is 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 ) = 1 − ( 0 ) 2 = 1 − 0 = 1
Calculating f(g(0)) Now that we have g ( 0 ) = 1 , we can find f ( g ( 0 )) by substituting g ( 0 ) = 1 into the expression for f ( x ) : f ( g ( 0 )) = f ( 1 ) = 3 ( 1 ) − 9 = 3 − 9 = − 6
Calculating f(0) Next, we need to find f ( 0 ) . We substitute x = 0 into the expression for f ( x ) : f ( 0 ) = 3 ( 0 ) − 9 = 0 − 9 = − 9
Calculating g(f(0)) Now that we have f ( 0 ) = − 9 , we can find g ( f ( 0 )) by substituting f ( 0 ) = − 9 into the expression for g ( x ) : g ( f ( 0 )) = g ( − 9 ) = 1 − ( − 9 ) 2 = 1 − 81 = − 80
Final Answer Therefore, f ( g ( 0 )) = − 6 and g ( f ( 0 )) = − 80 .
Examples
Function composition is a fundamental concept in mathematics with numerous real-world applications. For instance, consider a store that applies a discount and then adds sales tax. If f ( x ) represents the discount function and g ( x ) represents the sales tax function, then f ( g ( x )) would calculate the final price after tax is applied to the discounted price. Understanding function composition allows businesses to model and optimize pricing strategies, manage costs, and analyze financial data effectively.
