Mathematics › Functions › Composite Function
Let $\mathrm{f}(x)=\mathrm{e}^x-x$ and $\mathrm{g}(x)=x^2-x, \forall x \in \mathrm{R}$, then the set of all…
Let $\mathrm{f}(x)=\mathrm{e}^x-x$ and $\mathrm{g}(x)=x^2-x, \forall x \in \mathrm{R}$, then the set of all $x \in \mathrm{R}$, where the function $\mathrm{h}(x)=(\mathrm{fog})(x)$ is increasing is
$\left[0, \frac{1}{2}\right] \cup[1, \infty)$ $\left[-1,-\frac{1}{2}\right] \cup\left[\frac{1}{2}, \infty\right)$ $[0, \infty)$ $\left[-\frac{1}{2}, 0\right] \cup[1, \infty)$ Solution
$\begin{aligned}
& \mathrm{h}(x)=(\mathrm{fog})(x) \\
& \Rightarrow \mathrm{h}(x)=\mathrm{f}\left(x^2-x\right) \\
& \Rightarrow \mathrm{h}(x)=\mathrm{e}^{x^2-x}-x^2+x \\
& \therefore \quad \mathrm{h}^{\prime}(x)=\mathrm{e}^{x^2-x}(2 x-1)-2 x+1 \\
& \Rightarrow \mathrm{h}^{\prime}(x)=\left(\mathrm{e}^{x^2-x}-1\right)(2 x-1)
\end{aligned}$
For function $\mathrm{h}(x)$ to be increasing,
$\begin{aligned}
& \mathrm{h}^{\prime}(x) \geq 0 \\
& \Rightarrow\left(\mathrm{e}^{x^2-x}-1\right)(2 x-1) \geq 0 \\
& \Rightarrow x \in\left[0, \frac{1}{2}\right] \cup[1, \infty)
\end{aligned}$
$\begin{array}{c|c|c|c}
\mathrm{h}^{\prime}- & + & - & + \\
\hline & & & \\
0 & \frac{1}{2} & 1
\end{array}$
Asked in: MHT CET 2023 (10 May Shift 2)
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