f (x) = | x - 1 |, f: R+ → R and g (x) = ex, g: [-1, ∞) → R. If the function fog (x) is defined, then its domain and range respectively are

Question

f (x) = | x - 1 |, f: R+ → R and g (x) = ex, g: [-1, ∞) → R. If the function fog (x) is defined, then its domain and range respectively are (A) (0, ∞) and [0, ∞) (B) [-1, ∞) and [0, ∞) (C) [-1, ∞) and [1-(1/e), ∞] (D) [-1, ∞) and [(1/e) -1, ∞]

Tardigrade Admin · Accepted Answer

f (x) | x 1| = begincases 1 -x, text 0 < x < 1 [2ex] x-1, text x ge 1 endcases g (x) = ex, x 3 ge -1 (fog)(x) = begincases 1 -g(x), text 0 < g(x) < 1 i.e. -1 le x < 0 [2ex] g(x)-1, text g(x) ge 1 i.e. 0 le x endcases = begincases 1-ex , text-1 ge x < 0 [2ex] ex - 1, textx ge 0 endcases ∴ domain = [-1, ∞) fog is decreasing in [-1, 0) and increasing in [0, ∞) fog(-1) = 1 - (1/e) and fog (0) = 0 As x → ∞, fog (x) → ∞, ∴ range = [0, ∞) ∴ x = (1/2)loge ((y/2-y))

Q. $f (x) = ∣ x - 1∣, f : R^{+} \to R$ and $g (x) = e^{x}, g : [- 1, \infty) \to R$ . If the function fog $(x)$ is defined, then its domain and range respectively are

$(0, \infty)$ and $[0, \infty)$

$[- 1, \infty)$ and $[0, \infty)$

$[- 1, \infty)$ and $[1 - \frac{1}{e}, \infty]$

$[- 1, \infty)$ and $[\frac{1}{e} - 1, \infty]$

Q. f(x)=∣x−1∣,f:R+→R and g(x)=ex,g:[−1,∞)→R. If the function fog (x) is defined, then its domain and range respectively are

(0,∞) and [0,∞)

[−1,∞) and [0,∞)

[−1,∞) and [1−e1​,∞]

[−1,∞) and [e1​−1,∞]

Q. $f (x) = ∣ x - 1∣, f : R^{+} \to R$ and $g (x) = e^{x}, g : [- 1, \infty) \to R$ . If the function fog $(x)$ is defined, then its domain and range respectively are

$(0, \infty)$ and $[0, \infty)$

$[- 1, \infty)$ and $[0, \infty)$

$[- 1, \infty)$ and $[1 - \frac{1}{e}, \infty]$

$[- 1, \infty)$ and $[\frac{1}{e} - 1, \infty]$