If f(x) = x^2 - 7x and g(x) = x + 3, then what is the minimum value of f(g(x)) - 3x?

(1) -15
(2) -20
(3) -16
(4) -12

The minimum value of f ( g ( x )) − 3 x is found to be -16. This is achieved by evaluating the function f ( g ( x )) and determining the vertex of the resulting quadratic. The correct multiple choice option is (3) -16.
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Answered by Anonymous | 2025-07-04

To find the minimum value of the expression f ( g ( x )) − 3 x , we first need to substitute g ( x ) into f ( x ) . Given:
f ( x ) = x 2 − 7 x
g ( x ) = x + 3
First, we find f ( g ( x )) :
f ( g ( x )) = f ( x + 3 ) = ( x + 3 ) 2 − 7 ( x + 3 )
Expanding ( x + 3 ) 2 , we get:
( x + 3 ) 2 = x 2 + 6 x + 9
Substituting back into f ( g ( x )) , we have:
f ( g ( x )) = x 2 + 6 x + 9 − 7 x − 21
Simplifying, this becomes:
f ( g ( x )) = x 2 − x − 12
Next, we need to subtract 3 x from f ( g ( x )) :
f ( g ( x )) − 3 x = x 2 − x − 12 − 3 x
Simplifying further:
f ( g ( x )) − 3 x = x 2 − 4 x − 12
This is a quadratic function of the form a x 2 + b x + c , where a = 1 , b = − 4 , and c = − 12 .
To find the minimum value of a quadratic function a x 2 + b x + c , we use the vertex formula x = − 2 a b ​ . Here, this gives:
x = − 201 − 4 ​ = 2
Substitute x = 2 back into the expression x 2 − 4 x − 12 :
( 2 ) 2 − 4 ( 2 ) − 12 = 4 − 8 − 12 = − 16
Therefore, the minimum value of f ( g ( x )) − 3 x is − 16 .
Thus, the correct answer is option (3) : -16.

Answered by SophiaElizab | 2025-07-06