Question

Let f(x)=
`\frac{x {}^(2) }{x {}^(2 ) - 1 }`

and g(x)=
`(1)/( √(x - 1) )`

a) Find the domains of f (x) and g(x).
b) Find f (g(x)) and describe its domain.​

Answer 1

Domain:

f(x) has a denominator, which can't be zero. So, its domain is given by

`x^2-1\\eq 0 \iff x^2\\eq 1 \iff x\\eq \pm 1`

g(x) has a denominator as well. Moreover, it has a root. So, the content of the root can't be negative:

`x-1\geq 0 \iff x \geq 1`

And the denominator can't be zero:

`√(x-1)\\eq 0 \iff x-1 \\eq 0 \iff x \\eq 1`

So, the domain is
`x>1`

Composition:

We have

`f(g(x))=(g^2(x))/(g^2(x)-1) = ((1)/(x-1))/((1)/(x-1)-1) = ((1)/(x-1))/((1-x+1)/(x-1)) = ((1)/(x-1))/((2-x)/(x-1))=(1)/(2-x)`

The domain of this function is

`2-x\\eq 0 \iff x\\eq 2`

But we also have to remember about the domain of g(x): if g(x) is undefined, we can't compute f(g(x))!

So, the domain of f(g(x)) is

`x>1\ \land x\\eq 2`