Question

Given the functions defined by h(x)=x2−3x and k(x)=(x−2)−−−−−−√ , find kh(x) and write the domain of kh in interval notation

Answer 1

`k(h(x)) = √(x^2 - 3x - 2)`

`x = (\infty, -0.56]\ u\ [3.56, \infty)`

Explanation:

Given

`h(x) = x^2 - 3x`

`k(x) = √(x - 2)`

Solving (a): k(h(x))

`k(x) = √(x - 2)`

Replace x with h(x)

`k(h(x)) = √(h(x) - 2)`

Substitute:
`h(x) = x^2 - 3x`

`k(h(x)) = √(x^2 - 3x - 2)`

Solving (b): The domain

For the function to be defined, the expression in the root must be greater than or equal to 0; i.e.

`x^2 - 3x - 2 \ge 0`

Solve for x;

Using a calculator, we have:

`x \le -0.56` and
`x \ge 3.56` --- approximated

So, the domain is:

`x = (\infty, -0.56]\ u\ [3.56, \infty)`