Let f (x) = ex – x and g(x) = x2 – x, ∀ x ∈ R. Then the set of all x ∈ R, where the function h(x) = (fog) (x) is increasing, is

Engineering

Mathematics

Monotonicity

Question

Let f (x) = e^x – x and g(x) = x² – x, ∀ x ∈ R. Then the set of all x ∈ R, where the function h(x) = (fog) (x) is increasing, is

[0, ∞)

$[\frac{- 1}{2}, 0] \cup [1, \infty)$

$[0, \frac{1}{2}] \cup [1, \infty)$

$[- 1, \frac{- 1}{2}] \cup [\frac{1}{2}, \infty)$

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h(x) = f(g(x))

$∴$ h'(x) = f'(g(x))g'(x) and f'(x) = e^x – 1

h’(x) = (e^g(x)–1) g’(x)

$h' (x) = (e^{x^{2} - x} - 1) (2 x - 1) \geq 0$

Case : 1

$e^{x^{2} - x} \leq 1 and 2 x - 1 \geq 0$

⇒ x ∈ [1, ∞) ...............(1)

Case : 2

$e^{x^{2} - x} \geq 1 and 2 x - 1 \geq 0$

⇒ x ∈ [1, ∞) ...............(2)

From (i) and (ii)

$x \in [0, \frac{1}{2}] \cup [1, \infty)$