Suppose a function f(x) has domain (-1,1). Find the domains of f((x+1)/(x-1))
(R)-1
Can you explain more Sunny?
To find the domain of f((x+1)/(x-1)), we need to determine the values of x that make the expression (x+1)/(x-1) defined.
First, we know that the denominator cannot be zero since division by zero is undefined. So, we need to find the values of x that satisfy the equation x-1 ≠ 0.
If we solve x-1 ≠ 0 for x, we have x ≠ 1. Therefore, the expression (x+1)/(x-1) is defined for all values of x except x = 1.
However, we still need to consider the original domain of f(x). Given that f(x) has a domain of (-1, 1), we need to determine the values of (x+1)/(x-1) that fall within this domain.
To do that, we need to find the values of x that satisfy (-1 < (x+1)/(x-1) < 1).
First, let's solve the inequality (x+1)/(x-1) > -1:
(x+1)/(x-1) > -1
Multiply both sides by (x-1) to get rid of the fraction:
x + 1 > - (x - 1)
x + 1 > -x + 1
Combine like terms:
2x > 0
Divide both sides by 2:
x > 0
Now, let's solve the inequality (x+1)/(x-1) < 1:
(x+1)/(x-1) < 1
Multiply both sides by (x-1) to get rid of the fraction:
x + 1 < x - 1
Combine like terms:
2 < 0
We can see that the inequality 2 < 0 has no solutions, so there are no values of x that satisfy (-1 < (x+1)/(x-1) < 1).
Therefore, the domain of f((x+1)/(x-1)) is the set of all real numbers except x = 1.