Question

What is the domain and range of `F(x) = 5/(3-cos2x)`?

Answer 1

The domain is `(-oo, oo)`. The range is `[5/4, 5/2]`.

Explanation:

Much like a rational function in `x`, we must ensure that the denominator is not `0`.

`3 - cos2x ne 0`

`3 ne cos2x`

Note that the range of the cosine function is `[-1, 1]` so this inequality always holds true. In other words, the denominator is never equal to `0` so the domain is `(-oo, oo)`.

`F` is a continuous function so it suffices to find the maximum and minimum value for the range. Notice that only the denominator is affected by `x`. In this case, we wish to have the greatest and lowest denominator possible to minimize and maximize `F`.

`cos2x in [color(red)(-1), color(blue)(1)]`

The greatest denominator is `3 - (color(red)(-1))` which gives a minimum value of:

`5/(3 - (color(red)(-1))) = 5/4`

The lowest denominator is `3 - (color(blue)1)` which gives a maximum value of:

`5/(3 - (color(blue)1)) = 5/2`

The range of `F` must therefore be `[5/4, 5/2]`.