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CAT 2021 Slot 3 — QA Question 3

Composite FunctionsEasy

If f(x) = x2 – 7x and g(x) = x + 3, then the minimum value of f(g(x)) – 3x is

Answer & solution

Correct answer: -16

  • A

    -12

  • -16

  • C

    -15

  • D

    20

Solution

Easy

Form the composite f(g(x))f(g(x)) by substituting g(x)=x+3g(x)=x+3 into ff, subtract 3x3x, and simplify to a single quadratic in xx. A quadratic with positive leading coefficient attains its minimum at the vertex x=−b2ax=-\dfrac{b}{2a}.

1

Build the composite and subtract 3x3x. Replace every xx in ff by g(x)=x+3g(x)=x+3.

f(g(x))−3x=(x+3)2−7(x+3)−3x⇒ =x2+6x+9−7x−21−3x(expand)⇒ =x2−4x−12\begin{aligned} &f(g(x))-3x=(x+3)^2-7(x+3)-3x\\ &\Rightarrow\ =x^2+6x+9-7x-21-3x \quad\text{(expand)}\\ &\Rightarrow\ =x^2-4x-12 \end{aligned}
2

Locate the vertex. For ax2+bx+cax^2+bx+c with a>0a>0 the minimum is at x=−b2ax=-\dfrac{b}{2a}. Here a=1, b=−4a=1,\ b=-4.

x=−−42(1)=2\begin{aligned} &x=-\frac{-4}{2(1)}=2 \end{aligned}
3

Evaluate at the vertex. Substitute x=2x=2 into the quadratic from step 1.

(2)2−4(2)−12=4−8−12⇒ =−16\begin{aligned} &(2)^2-4(2)-12=4-8-12\\ &\Rightarrow\ =-16 \end{aligned}
Minimum value=−16\text{Minimum value}=-16

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CAT 2021 Slot 3 QA Q3 — Composite Functions solution | TheCATExam