Find domain and Range of function f which is defined by. f(x) = 1/2-cos3x.

Question

Find domain and Range of function f which is defined by.

f(x) = \(\frac{1}{2-cos3x}\)

Answer 1

Given, function f(x) =  f(x) = \(\frac{1}{2-cos3x}\)

We know that for real x, value of cos 3x lies between -1 and 1.

∴ - 1 ≤ cos 3x ≤ 1, “x ∈ R

According to definition of function

2 - cos 3x ≠ 0, “x ∈ R

∴ Function is defined for ∀ x ∈ R

∴ Domain of f = R

Again, maximum value of cos 3x is

∴ f(x) maximum value of f(x) is = 1

and minimum value of cos3x is - 1

∴ Minimum value of f(x) = \(\frac{1}{2-(-1)} = \frac{1}{3}\) 

Thus, Range of f = \([\frac{1}{3},1]\)