TS EAMCET · Maths · Functions
If \(\mathrm{D} \subseteq \mathbb{R}\) and \(f: \mathrm{D} \rightarrow \mathbb{R}\) defined by \(f(x)=\frac{x^2+x+a}{x^2-x+a}\) is a surjection then ' \(a\) ' lies in the interval
- A \(\mathbb{R}\)
- B \((0, \infty)\)
- C \((-\infty, 0)\)
- D \((0,1)\)
Answer & Solution
Correct Answer
(C) \((-\infty, 0)\)
Step-by-step Solution
Detailed explanation
\((y-1)x^2 - (y+1)x + a(y-1) = 0\) For \(f\) to be a surjection, for every \(y \in \mathbb{R}\), a real \(x\) must exist. If \(y=1\), \(-2x=0 \Rightarrow x=0\). For \(f(0)=1\) to be defined, \(a \neq 0\). If \(y \neq 1\), the discriminant must be non-negative:…
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