If [tex]f(a)=6 a-7[/tex] and [tex]g(a)=a^2[/tex], then [tex]g(f(a))=[/tex]
A. [tex]36 a^2+49[/tex]
B. [tex]36 a^2-84 a+49[/tex]
C. [tex]36 a^2-42 a+49[/tex]
D. [tex]6 a^2-7[/tex]
E. [tex]a^2(6 a-7)[/tex]
Answer (1)
Substitute f ( a ) into g ( a ) : g ( f ( a )) = ( f ( a ) ) 2 .
Replace f ( a ) with 6 a − 7 : g ( f ( a )) = ( 6 a − 7 ) 2 .
Expand the expression: ( 6 a − 7 ) 2 = 36 a 2 − 84 a + 49 .
The final answer is 36 a 2 − 84 a + 49 .
Explanation
Understanding the Problem We are given two functions, f ( a ) = 6 a − 7 and g ( a ) = a 2 . We want to find the composite function g ( f ( a )) , which means we need to substitute f ( a ) into g ( a ) .
Substituting f(a) into g(a) To find g ( f ( a )) , we replace the input 'a' in the function g ( a ) with the entire function f ( a ) . So, we have g ( f ( a )) = ( f ( a ) ) 2 . Since f ( a ) = 6 a − 7 , we get g ( f ( a )) = ( 6 a − 7 ) 2 .
Expanding the Expression Now we need to expand the expression ( 6 a − 7 ) 2 . Recall that ( x − y ) 2 = x 2 − 2 x y + y 2 . Applying this to our expression, we have ( 6 a − 7 ) 2 = ( 6 a ) 2 − 2 ( 6 a ) ( 7 ) + ( 7 ) 2 .
Simplifying the terms Let's simplify the expression: ( 6 a ) 2 = 36 a 2 2 ( 6 a ) ( 7 ) = 84 a ( 7 ) 2 = 49 So, ( 6 a − 7 ) 2 = 36 a 2 − 84 a + 49 .
Final Answer Therefore, g ( f ( a )) = 36 a 2 − 84 a + 49 . Looking at the multiple-choice options, we see that this matches option b.
Examples
Composite functions are useful in many real-world scenarios. 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 price after the discount and g ( x ) = 1.05 x represents the price after tax, then g ( f ( x )) would give the final price of an item after both the discount and the tax are applied. Understanding composite functions helps in modeling such sequential operations.
