Question

The range of the function  \[f\left( x \right) = \frac{x^2 - x}{x^2 + 2x}\]  is 

 

विकल्प

• (a) R

• (b) R − {1}

• (c) R − {−1/2, 1}

• (d) None of these

   

Answer 1

(c) R − {−1/2, 1}

\[f\left( x \right) = \frac{x^2 - x}{x^2 + 2x}\]

\[\text{ Let }  y = \frac{x^2 - x}{x^2 + 2x} \left[\text{  Also,}  x \neq 0 \right]\]
\[ \Rightarrow y = \frac{x(x - 1)}{x(x + 2)}\]
\[ \Rightarrow y = \frac{(x - 1)}{(x + 2)}\]
\[ \Rightarrow xy + 2y = x - 1\]
\[ \Rightarrow x = \frac{2y + 1}{1 - y}\]
\[\text{ Here } , 1 - y \neq 0 . \]
\[\text{ or } , y \neq 1 . \]
\[\text{ Also } , x \neq 0\]
\[ \Rightarrow \frac{2y + 1}{1 - y} \neq 0\]
\[ \Rightarrow y \neq - \frac{1}{2}\]
\[\text{ Thus, range  } (f) = R - { - \frac{1}{2}, 1} . \]