Find the ranges of the following functions: [vii] f ( x ) = e x + e - x 2

Given - f x = e x + e - x 2 Let f x = y ⇒ y = e x + e - x 2 ⇒ 2 y = e x + 1 e x ⇒ 2 y e x = e 2 x + 1 ⇒ e 2 x - 2 y e x + 1 = 0 Now it is quadratic in e x ⇒ e x = 2 y ± 4 y 2 - 4 2 This is defined only if 4 y 2 - 4 ≥ 0 ⇒ 4 y 2 ≥ 4 Divide both sides by 4 ⇒ y 2 ≥ 1 ⇒ y ≥ 1 Therefore, the range of this function is [ 1 , ∞ ) .

Find the ranges of the following functions: [vii] f ( x ) = e x + e - x 2

Given - f x = e x + e - x 2 Let f x = y ⇒ y = e x + e - x 2 ⇒ 2 y = e x + 1 e x ⇒ 2 y e x = e 2 x + 1 ⇒ e 2 x - 2 y e x + 1 = 0 Now it is quadratic in e x ⇒ e x = 2 y ± 4 y 2 - 4 2 This is defined only if 4 y 2 - 4 ≥ 0 ⇒ 4 y 2 ≥ 4 Divide both sides by 4 ⇒ y 2 ≥ 1 ⇒ y ≥ 1 Therefore, the range of this function is [ 1 , ∞ ) .

The domain of the definition of the function f x = 1 4 - x 2 + l o g 10 x 3 - x is:

- 2 , - 1 ∪ - 1 , 0 ∪ 2 , ∞

If f x + 2 f 1 x = 3 x , x ≠ 0 , and S = x ∈ R : f x = f - x , then S

Contains more than two elements

HARD

Earn 100

Find the ranges of the following functions:
[vii] $f (x) = \frac{e^{x} + e^{- x}}{2}$

Important Questions on Functions

EASY

Mathematics>Differential Calculus>Functions>Basics of Functions

The domain of the definition of the function

f (x) = \frac{1}{4 - x^{2}} + {l o g}_{10} (x^{3} - x)

is:

EASY

Mathematics>Differential Calculus>Functions>Basics of Functions

Let

f : R \to R

satisfy

f (x) f (y) = f (x y)

for all real numbers

x

and

y

.

f (2) = 4

,

then

f (\frac{1}{2}) =

MEDIUM

Mathematics>Differential Calculus>Functions>Basics of Functions

Let

S

be the set of points where the function

, f (x) = |2 - | x - 3 ||, x \in R,

is not differentiable. Then

\sum_{x \in S} f (f (x))

is equal to

EASY

Mathematics>Differential Calculus>Functions>Basics of Functions

Let

f : [2, \infty) \to R

be the function defined by

f (x) = x^{2} - 4 x + 5

, then the range

HARD

Mathematics>Differential Calculus>Functions>Basics of Functions

The range of the polynomial

P (x) = 4 x^{3} - 3 x

x

varies over the interval

(- \frac{1}{2}, \frac{1}{2})

HARD

Mathematics>Differential Calculus>Functions>Basics of Functions

Let

f : (1, 3) \to R

be a function defined by

f (x) = \frac{x [x]}{1 + x^{2}},

where

[x]

denotes the greatest integer

\leq x .

Then the range of

f

MEDIUM

Mathematics>Differential Calculus>Functions>Basics of Functions

f (x) + 2 f (\frac{1}{x}) = 3 x, x \neq 0,

and

S = \{x \in R : f (x) = f (- x)\}

, then

S

EASY

Mathematics>Differential Calculus>Functions>Basics of Functions

Let

f : R \to R

be defined by

f (x) = \{\begin{cases} 2 x; & x > 3 \\ x^{2}; & 1 < x \leq 3 \\ 3 x; & x \leq 1 \end{cases}

Then

f (- 1) + f (2) + f (4)

HARD

Mathematics>Differential Calculus>Functions>Basics of Functions

Let

a, b, c \in R

. If

f (x) = a x^{2} + b x + c

is such that

a + b + c = 3

and

f (x + y) = f (x) + f (y) + x y, \forall x, y \in R

, then

\sum_{n = 1}^{10} f (n)

is equal to:

EASY

Mathematics>Differential Calculus>Functions>Basics of Functions

Let

f : (- 1, 1) \to (- 1, 1)

be continuous,

f (x) = f (x^{2})

for all

x \in (- 1, 1)

and

f (0) = \frac{1}{2}

,

then the value of

4 f (\frac{1}{4})

MEDIUM

Mathematics>Differential Calculus>Functions>Basics of Functions

The domain of the function $f : R \to R$ defined by $f (x) = \sqrt{x^{2} - 7 x + 12}$ is

MEDIUM

Mathematics>Differential Calculus>Functions>Basics of Functions

Let

f : [0,1] \to R

be such that

f (x y) = f (x) \cdot f (y),

for all

x, y \in [0,1],

and

f (0) \neq 0 .

y = y (x)

satisfies the differential equation,

\frac{d y}{d x} = f (x)

with

y (0) = 1

then

y (\frac{1}{4}) + y (\frac{3}{4})

is equal to:

EASY

Mathematics>Differential Calculus>Functions>Basics of Functions

The domain of the function

f

given by

f (x) = \sqrt{x - 1}

HARD

Mathematics>Differential Calculus>Functions>Basics of Functions

Let

\sum_{k = 1}^{10} f (a + k) = 16 (2^{10} - 1),

where the function

f

satisfies

f (x + y) = f (x) f (y)

for all natural numbers

x, y

and

f (1) = 2 .

Then the natural number

a

is:

MEDIUM

Mathematics>Differential Calculus>Functions>Basics of Functions

Let

f : R \to R

be a function which satisfies

f (x + y) = f (x) + f (y), \forall x, y \in R

. If

f (1) = 2

and

g (n) = \sum_{k = 1}^{(n - 1)} f (k),

n \in N

then the value of

n

, for which

g (n) = 20

, is

EASY

Mathematics>Differential Calculus>Functions>Basics of Functions

The domain of the function

f (x) = \sqrt{\frac{1}{|x - 2| - (x - 2)}}

HARD

Mathematics>Differential Calculus>Functions>Basics of Functions

Let the function $f : [0, 1] \to ℝ$ be defined by $f (x) = \frac{4^{x}}{4^{x} + 2}$ . Then the value of $f (\frac{1}{40}) + f (\frac{2}{40}) + f (\frac{3}{40}) + \dots + f (\frac{39}{40}) - f (\frac{1}{2})$ is_________

MEDIUM

Mathematics>Differential Calculus>Functions>Basics of Functions

Let

f : x \to y

be such that

f (1) = 2

and

f (x + y) = f (x) f (y)

for all natural numbers

x

and

y

.

\sum_{k = 1}^{n} f (a + k) = 16 (2^{n} - 1)

then

a

is equal to

HARD

Mathematics>Differential Calculus>Functions>Basics of Functions

Let

f

be a continuous function defined on

[0, 1]

such that

\int_{0}^{1} f^{2} (x) d x = {(\int_{0}^{1} f (x) d x)}^{2}

. Then the range of

f

MEDIUM

Mathematics>Differential Calculus>Functions>Basics of Functions

Let,

f : R \to R

be a function such that

f (x) = x^{3} + x^{2} f^{'} (1) + x f^{"} (2) + f^{'''} (3), \forall x \in R .

Then

f (2)

equals

Find the ranges of the following functions: [vii] f(x)=ex+e-x2

Important Questions on Functions

Learn and Prepare for any exam you want !

Find the ranges of the following functions:
[vii] $f (x) = \frac{e^{x} + e^{- x}}{2}$