Question

Let `f( x ) = sqrt(− 3−x)` and `g ( x ) = x^2 − 4x` `f@g=`? The domain of `f@g` is?

Answer 1

`f(g(x)) = sqrt(−x^2 + 4x-3)`

The domain is `1 <= x <= 3`

Explanation:

Given: `f(x) = sqrt(− 3−x)` and `g(x) = x^2 − 4x`

Start with `f(x)`:

`f(x) = sqrt(− 3−x)`

Substitute `g(x)` everywhere you see an ex:

`f(g(x)) = sqrt(− 3−g(x))`

One the right side, substitute `g(x) = x^2 − 4x`:

`f(g(x)) = sqrt(− 3−(x^2 − 4x))`

Distribute the minus sign:

`f(g(x)) = sqrt(− 3−x^2 + 4x)`

Arrange the powers in descending order:

`f(g(x)) = sqrt(−x^2 + 4x-3)`

To find the domain, we must specify:

`-x^2+4x-3 >=0`

Multiply both sides by -1:

`x^2-4x+3 <= 0`

A quadratic of this type will be less than 0 between the two roots, therefore, we shall find the roots by changing the inequality to an equation:

`x^2-4x+3 = 0`

Factor:

`(x-1)(x-3)= 0`

`x = 1 and x = 3`

Please observe that the original inequality, `-x^2+4x-3 >=0`, are true between the two roots inclusively.

graph{-x^2+4x-3 [-0.53, 4.336, -1.124, 1.31]}

This makes the domain `1 <= x <= 3`