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The Only Integer Missing from the Range of (x^2-3x+2)/(x^2+x-6) | JEE

JEE Maths question with a full step-by-step solution.

Question

Find the only integer which does not lie in the range of the function

f(x) = \frac{x^{2}-3x+2}{x^{2}+x-6} .

Solution

Answer: 1

Step 1: Factorise the numerator and the denominator.

x^{2}-3x+2 = (x-1)(x-2), \qquad x^{2}+x-6 = (x+3)(x-2) .

Step 2: Note the domain before cancelling. The denominator vanishes at

x = -3

and

x = 2

, so

x \ne -3, \qquad x \ne 2 .

Step 3: Cancel the common factor, remembering it is only legal because

x \ne 2

f(x) = \frac{x-1}{x+3}, \qquad x \ne -3,\ x \ne 2 .

Step 4: Find the range of

\dfrac{x-1}{x+3}

by making

x

the subject. Put

y = \dfrac{x-1}{x+3}

y(x+3) = x - 1 \implies x(y-1) = -1 - 3y \implies x = \frac{1+3y}{1-y} .

This has a solution for every

y

except

y = 1

, so

y = 1

is not attained. Step 5: Remove the value lost at the cancelled point

x = 2

\frac{2-1}{2+3} = \frac15 ,

y = \dfrac15

is also missing from the range (that is the "hole" in the graph). Step 6: Write the range.

\text{Range} = \mathbb{R} - \left\{1,\ \frac15\right\} .

Step 7: Pick out the integers. Of the two excluded values, only

1

is an integer. Answer:

1

(i.e.

1.00

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