Let the function f be defined in (0,1). Two more functions are given as g ( x )= e x and h ( x )= ln | x | Domain of f ( g ( x ))+ f ( h ( x )) is

Question

Let the function f be defined in (0,1). Two more functions are given as g ( x )= e x and h ( x )= ln | x | Domain of f ( g ( x ))+ f ( h ( x )) is (A) (-e,-1) (B) (- e ,-1) ∪(1, e ) (C) [- e ,-1] (D) (-∞, 0)

Tardigrade Admin · Accepted Answer

f ( g ( x ))+ f ( h ( x ))= f ( e x )+ f ( ln | x |) ∵ f text is defined in (0,1) ∴ 0< e x <1 text and 0< ln | x |<1 x ∈(-∞, 0) text and 1<| x |< e x ∈(- e ,-1) ∪(1, e ) ∴ text Domian is (- e ,-1)

Q. Let the function $f$ be defined in $(0, 1)$ . Two more functions are given as $g (x) = e^{x}$ and $h (x) = ln ∣ x ∣$
Domain of $f (g (x)) + f (h (x))$ is

$(- e, - 1)$

$(- e, - 1) \cup (1, e)$

$[- e, - 1]$

$(- \infty, 0)$

$f (g (x)) + f (h (x)) = f (e^{x}) + f (ln ∣ x ∣)$
$∵ f is defined in (0, 1)$
$∴ 0 < e^{x} < 1 and 0 < ln ∣ x ∣ < 1$
$x \in (- \infty, 0) and 1 < ∣ x ∣ < e$
$x \in (- e, - 1) \cup (1, e)$
$∴ Domian is (- e, - 1)$

Q. Let the function f be defined in (0,1). Two more functions are given as g(x)=ex and h(x)=ln∣x∣ Domain of f(g(x))+f(h(x)) is

(−e,−1)

(−e,−1)∪(1,e)

[−e,−1]

(−∞,0)