Let f(x) = ex -- x and g(x) = x2 -- x, [ ] x [ ] R.

Question

Let f(x) = ex -- x and g(x) = x2 -- x, $$\forall $$ x $$ \in $$ R. Then the set of all x $$ \in $$ R, where the function h(x) = (fog) (x) is increasing, is :

Accepted Answer

f(x) = ex -- x, g(x) = x2 -- x

f(g(x)) = e(x2 - x) - (x2 - x)

If f(g(x)) is increasing function

(f (g(x)))x = $${e^{\left( {{x^2} - x} \right)}} \times (2x - 1) - 2x + 1$$

$$ \Rightarrow \mathop {(2x - 1)}\limits_A \mathop {[{e^{\left( {{x^2} - x} \right)}} - 1]}\limits_B $$

A \& B are either both positive or negative

for (f (g(x)))' $$ \ge $$ 0,

$$x \in \left[ {0,{1 \over 2}} \right] \cup \left[ {1,\infty } \right)$$

Solutions · 1