First slide

Functions (XII)

Question

The value of the function $f (x) = \frac{x^{2} - 3 x + 2}{x^{2} + x - 6}$ lies in the interval

Easy

A

$(- \infty, \infty) \ \{\frac{1}{5}, 1\}$
B

(–∞, ∞)
C

(–∞, ∞)\{1}
D

None of these

Solution

f (x) is defined if x² + x – 6 ≠ 0
i.e., (x + 3) (x – 2) ≠ 0 i.e. x ≠ – 3, 2
$∴ Domain (f) = (- \infty, \infty) \ {- 3, 2}$
$Let y = \frac{x^{2} - 3 x + 2}{x^{2} + x - 6} \begin{array}{l} \Rightarrow x^{2} y + xy - 6 y = x^{2} - 3 x + 2 \\ \Rightarrow x^{2} (y - 1) + x (y + 3) - (6 y + 2) = 0 \end{array}$
For x to be real, (y + 3)² + 4 (y – 1) (6y + 2) ≥ 0
⇒ 25y² – 10y + 1 ≥ 0 i.e. (5y – 1)² ≥ 0
which is true for all real y.
$∴$ Range of f = (–∞, ∞).

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App