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.

Question

asked Oct 21, 2020 182k views

1 Answer

Jan Schaefer · Answer 1 · 2020-10-26T02:47:28+0000

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

$x>1\ \land x\\eq 2$