Test: Domain and Range - Grade 12 MCQ

Domain is the set of all the numbers on which a function is defined. It may be real as well.

A square root function is not defined for negative real numbers.

Range is the set of all values which a function may take.

Function x^-1 is not defined for x=0, otherwise it defined for every real number.

Function x^-1 may take any real number hence it’s range is all real numbers.

The function cannot take negative values,hence range is (0, ∞).

Since minimum value of x² is 0, thus x² +4 may take any value between [4,∞).

∵ Identity functions maps each element onto itself, i.e. I(x) = x, i.e., the domain and range are equal.

If sinθ = x ⇒ θ = sin^-1x, for θ ∈ [-π/2, π/2]
sin^-1 (sin x) =x for -π /2 ≤ x ≤ π/2
Given that, f(x) = sin^-1
We know that sin^-1x is defined for x ∈ [-1, 1]
∴ f(x) = sin^-1is defined for 

If cosθ = x ⇒ θ = cos-1x, for θ ∈ [-π/2, π/2]
We have, y = cos–1(x2 – 4)
⇒ cos y = x² - 4
Now -1 ≤ cosθ  ≤ 1 ∀ θ ∈ R
∴ -1 ≤ x2 - 4 ≤ 1
⇒ 3 ≤ x² ≤ 5
⇒ √3 ≤ |x| ≤ √5
⇒ x ∈ [−√5, −√3] ∪ [√3, √5]
∴  Domain of y = cos–1(x2 – 4) is [−√5, −√3] ∪ [√3, √5]