Find the range of the following functions given by
`f(x)=(3)/(2-x^(2)) " "` (ii) `f(x)=1-|x-2|`
(iii) `f(x)=|x-3| " "` (iv) `f(x)=1+3 cos 2x`

Explanation

Let's solve the problem step by step for each function to find their ranges.

### (i) \( f(x) = \frac{3}{2 - x^2} \)

1. **Identify the domain**: The function is defined for all \( x \) such that \( 2 - x^2 \neq 0 \). This gives us \( x^2 \neq 2 \), or \( x \neq \pm \sqrt{2} \).

2. **Analyze the function**: As \( x^2 \) increases, \( 2 - x^2 \) decreases. Therefore, \( f(x) \) will approach \( 0 \) as \( x^2 \) approaches \( 2 \) from either side, and \( f(x) \) approaches \( \infty \) as \( x^2 \) approaches \( 2 \) from below.

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