Question

The domain of the function f(x) = `(cos^-1((x^2  -  5x  +  6)/(x^2  -  9)))/(log_e(x^2 - 3x + 2)` is ______.

Options

• (–∞, 1) ∪ (2, ∞)

• (2, ∞)

• `[-1/2, 1) ∪ (2, ∞)`

• `[-1/2, 1) ∪ (2, ∞) - {(3 + sqrt(5))/2, (3 - sqrt(5))/2}`

Answer 1

The domain of the function f(x) = `(cos^-1((x^2  -  5x  +  6)/(x^2  -  9)))/(log_e(x^2 - 3x + 2)` is `underlinebb([-1/2, 1) ∪ (2, ∞) - {(3 + sqrt(5))/2, (3 - sqrt(5))/2}`.

Explanation:

Given function domain is [–1, 1].

`-1 ≤ (x^2 - 5x + 6)/(x^2 - 9) ≤ 1`

Take the maximum value and subtract 1 from both sides.

`(x^2 - 5x + 6)/(x^2 - 9) - 1 ≤ 0, 1/(x + 3) ≥ 0`

x ∈ (–3, ∞)  ...(i)

Take minimum value and add 1 on both sides.

`(x^2 - 5x + 6)/(x^2 - 9) + 1 ≥ 0, (2x + 1)/(x + 3) ≥ 0`

`x ∈ (-∞, -3) ∪ [-1/2, ∞)`  ...(ii)

Now, take the intersection of two equations (i) and (ii)

`x ∈ [-1/2, ∞) `

Now, take x² – 3x + 2 > 0, x ∈ (–∞, 1) ∪ (2, ∞)

`x^2 - 3x + 2 ≠ 1, x ≠ (3 ± sqrt(5))/2` 

Take intersection of all the solutions.

`[-1/2, 1) ∪ (2, ∞) - {(3 + sqrt(5))/2, (3 - sqrt(5))/2}`