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CAT 2021 Slot 2 — QA Question 22

Graph & Maximum or Minimum value of Quadratic functionMedium

For all real values of x, the range of the function f(x) = x2+2x+42x2+4x+9 is

Answer & solution

  • [37,12)

  • B

    [37,89)

  • C

    [49,12]

  • D

    (37,12)

Solution

Medium

Rewrite the fraction so the numerator is "half the denominator minus a small piece". This turns f(x)f(x) into 12(1−12x2+4x+9)\tfrac12\Big(1-\tfrac{1}{2x^{2}+4x+9}\Big), whose behaviour is governed entirely by the quadratic 2x2+4x+92x^{2}+4x+9. Find that quadratic's minimum (it has no maximum, →∞\to\infty) to get the range of ff.

1

Manipulate into a clean form. Make the numerator track the denominator: 2(x2+2x+4)=2x2+4x+8=(2x2+4x+9)−12(x^{2}+2x+4)=2x^{2}+4x+8=(2x^{2}+4x+9)-1.

f(x)=x2+2x+42x2+4x+9=12⋅2x2+4x+82x2+4x+9⇒ f(x)=12((2x2+4x+9)−12x2+4x+9)=12(1−12x2+4x+9)\begin{aligned} &f(x)=\frac{x^{2}+2x+4}{2x^{2}+4x+9}=\frac{1}{2}\cdot\frac{2x^{2}+4x+8}{2x^{2}+4x+9}\\ &\Rightarrow\ f(x)=\frac{1}{2}\left(\frac{(2x^{2}+4x+9)-1}{2x^{2}+4x+9}\right)=\frac{1}{2}\left(1-\frac{1}{2x^{2}+4x+9}\right) \end{aligned}
2

Range of the inner quadratic g(x)=2x2+4x+9g(x)=2x^{2}+4x+9. Upward parabola, minimum at x=−42⋅2=−1x=-\tfrac{4}{2\cdot 2}=-1.

g(−1)=2(1)+4(−1)+9=7⇒ g(x)∈[7,∞)\begin{aligned} &g(-1)=2(1)+4(-1)+9=7\\ &\Rightarrow\ g(x)\in[7,\infty) \end{aligned}
3

Push through to ff. As gg ranges over [7,∞)[7,\infty), 1g\tfrac{1}{g} ranges over (0,17](0,\tfrac17], so 1−1g1-\tfrac1g ranges over [67,1)[\tfrac67,1), and halving gives ff.

Min: f=12(1−17)=12⋅67=37(attained at x=−1)Sup: f→12(1−0)=12(as g→∞, never reached)⇒ Range=[37, 12)\begin{aligned} &\text{Min: } f=\frac{1}{2}\left(1-\frac{1}{7}\right)=\frac{1}{2}\cdot\frac{6}{7}=\frac{3}{7}\quad\text{(attained at } x=-1)\\ &\text{Sup: } f\to\frac{1}{2}(1-0)=\frac{1}{2}\quad\text{(as } g\to\infty,\text{ never reached)}\\ &\Rightarrow\ \text{Range}=\left[\frac{3}{7},\ \frac{1}{2}\right) \end{aligned}
Range=[37, 12)\text{Range}=\left[\dfrac{3}{7},\ \dfrac{1}{2}\right)
CAT 2021 Slot 2 QA Q22: For all real values of x, the range of the function f(x) = x 2 + 2 x + 4 2 x 2 + 4 x + 9 is — Solution | TheCATExam