The range of $f(x)=4\sin^{-1}\left(\frac{x^{2}}{x^{2}+1}\right)$ is

JEE Main · 2023 · 13 Apr 2023 · Shift 2 · Inverse Trigonometric Functions

The range of

f (x) = 4 sin^{- 1} (\frac{x ^{2}}{x ^{2} + 1})

Answer: (B)

Let

t = \frac{x ^{2}}{x ^{2} + 1} = 1 - \frac{1}{x ^{2} + 1}

.

Since

x^{2} \geq 0

for all real

x

, we have

x^{2} + 1 \geq 1

, so

\frac{1}{x ^{2} + 1} \in (0, 1]

.

Therefore,

t = 1 - \frac{1}{x ^{2} + 1} \in [0, 1)

.

The range of

sin^{- 1} (t)

for

t \in [0, 1)

[0, \frac{π}{2})

.

Therefore, the range of

f (x) = 4 sin^{- 1} (t)

[0, 2 π)

JEE Main · 2023 · 13 Apr 2023 · Shift 2 · Single correct option

If the domain of the function $f (x) = cos^{- 1} (\frac{2 x - 5}{11 - 3 x}) + sin^{- 1} (2 x^{2} - 3 x + 1)$ is the interval $[α, β]$ , then $α + 2 β$ is equal to :
JEE Main 2026View solution →
The number of solutions of $tan^{- 1} 4 x + tan^{- 1} 6 x = \frac{π}{6}$ , where $- \frac{1}{2 6} < x < \frac{1}{2 6}$ is equal to
JEE Main 2026View solution →
If the domain of the function $f (x) = sin^{- 1} (\frac{1}{x ^{2} - 2 x - 2})$ , is $(- \infty, α] \cup [β, γ] \cup [δ, \infty)$ , then $α + β + γ + δ$ is equal to
JEE Main 2026View solution →
If $k = tan (\frac{π}{4} + \frac{1}{2} cos^{- 1} (\frac{2}{3})) + tan (\frac{1}{2} sin^{- 1} (\frac{2}{3}))$ then the number of solutions of the equation …
JEE Main 2026View solution →
Considering the principal values of inverse trigonometric functions, the value of the expression $tan (2 sin^{- 1} (\frac{2}{13}) - 2 cos^{- 1} (\frac{3}{10}))$ is equal to :
JEE Main 2026View solution →
Let [ • ] denote the greatest integer function. If the domain of the function $f (x) = cos^{- 1} (\frac{4 x + 2 [ x ]}{3})$ is $[α, β]$ , then $12 (α + β)$ is equal to :
JEE Main 2026View solution →