First slide

Functions (XII)

Question

The range of a function y = f (x) is the set of all possible output values f (x) corresponding to every input x in the domain of f and is denoted as f (A) if A is the domain. For finding the range of a function y = f (x), first of all, find the domain of f.
If the domain consists of finite number of points, then the range consists of set of corresponding f (x) values. If the domain consists of whole real line or real line minus some finite points, then express x in terms of y as x = g(y). The values of y for which g is defined is the required range.
If the domain is a finite interval, find the intervals in which f (x) increases/decreases and then find the extreme values of the function in those intervals. The union of those intervals is the required range.

Moderate

Question

If e^x + e^f(x) = e, then range of the function f is

A

(–∞, 1]
B

(–∞, 1)
C

(1, ∞)
D

[1, ∞)

Solution

We have,
e^x + e^f(x) = e ⇒ e^f(x) = e – e^x
⇒ f (x) = log (e – e^x)
For f (x) to be defined, e – e^x > 0
⇒ e¹> e^x ⇒ x < 1
$∴$ Domain of f = (– ∞, 1)
Let y = log (e – e^x) ⇒ e^y = e – e^x
⇒ e^x = e – e^y
⇒ x = log (e – e^y)
For x to be real, e – e^y > 0
⇒ e¹ > e^y ⇒ y < 1
$∴$ Range of f = (– ∞, 1)

Question

The range of the function $f (x) = \frac{\sin (π [x^{2} + 1])}{x^{4} + 1}$ . where [⋅] denotes the greatest integer function, is

A

[0, 1]
B

[–1, 1]
C

{0}
D

None of these

Solution

Since [x² + 1] is an integer,
$\begin{array}{l} ∴ \sin (π [x^{2} + 1]) = 0 \\ \Rightarrow f (x) = \frac{\sin (π [x^{2} + 1])}{x^{4} + 1} = 0 \end{array}$
Hence, Range of f = R_f = {0}

Question

Range of values of f (x) = 1 + sinx + sin³x + sin⁵x+….., x ∈ $(- \frac{π}{2}, \frac{π}{2})$ is

A

(0, 1)
B

(– ∞, ∞)
C

(– 2, 2)
D

None of these

Solution

We have, $f (x) = 1 + \frac{\sin x}{\cos^{2} x}$
$\begin{array}{l} \Rightarrow f^{'} (x) = \frac{\cos^{2} x (\cos x) + \sin x (2 \cos xsin x)}{\cos^{4} x} \\ = \frac{\cos x (\cos^{2} x + 2 \sin^{2} x)}{\cos^{4} x} = \frac{1 + \sin^{2} x}{\cos^{3} x} \end{array}$
⇒ f ′(x) > 0.
$∴$ f (x) is increasing function.
$lim_{x \to \frac{- π}{2}} (1 + \frac{\sin x}{\cos^{2} x}) = - \infty$
$and, lim_{x \to \frac{π}{2}} (1 + \frac{\sin x}{\cos^{2} x}) = \infty$
$∴$ Range = (–∞, ∞)

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App