Question

Find the domain of g(x) = cos^–1 (x² – 1). Hence, find the value of x for which g(x) = `pi/3`.

Also, write the range of cos^–1 x other than its principal branch.

Answer 1

Given, 

g(x) = cos^–1 (x² – 1) 

cos^–1 : [–1 1] → [0, π] 

–1 ≤ x² –1 ≤ 1 

adding ‘1’ both sides,

0 ≤ x² ≤ 2

Hence, x² ≥ 2 ∀x∈R

The domain `-sqrt2≤x≤sqrt2`

`{x∈R|-sqrt2≤x≤sqrt2}`

Now, g(x) = `pi/3`

⇒ `cos^(–1) (x^2 -1) = pi/3`

⇒ `(x^2-1) = cospi/3`

⇒ `(x^2-1) = 1/2`

⇒ `x^2 = 1+1/2`

⇒ `x^2 = 3/2`

⇒ x = `±sqrt(3/2)`

⇒ x = `±sqrt6/2`

`sqrt2≈1.414;sqrt6/2≈2.449/2`

 ⇒ ≈1.225

Range of cos^–1x other than principal value branch

{[nπ, (n + 1)π]; n ∈ 2}