Question

Let $f(x)$ and $g(x)$ be two continuous functions defined from $R \rightarrow R$, such that $f\left(x_{1}\right) > f\left(x_{2}\right)$ and $g\left(x_{1}\right) > g\left(x_{2}\right)$ $\forall x_{1} > x_{2}$. Then the solution set of $f\left(g\left(\alpha^{2}-2 \alpha\right)\right) > f(g(3 \alpha-4))$, is

• A. $\alpha \in[1, 4)$
• B. $\alpha \in(1, 4)$
• C. $\alpha \in(1, 4]$
• D. None of these

Answer 1

Answer: (B)

Obviosuly, $f$ is increasing and $g$ is decreasing in $R$.
Hence, $f\left(g\left(\alpha^{2}-2 \alpha\right)\right) > f(g(3 \alpha-4))$
or $g\left(a^{2}-2 \alpha\right) > g(3 \alpha-4) $
$(\because f$ is increasing $)$
or $\alpha^{2}-2 \alpha < 3 \alpha-4$
(As $g$ is decreasing)
or $\alpha^{2}-5 \alpha+4 < 0$ or
$(\alpha-1)(\alpha-4) < 0$ or $\alpha \in(1,4)$