For f(x)=(1-x)/(1+x) and g(x)=x/(1-x) , find the simplified form for f [g(x)] and state the domain
Damon and Reiny are both correct about getting to "1 - 2x", but after that they're both utter morons.
You're looking for the domain (x), right? That means you start with all real numbers by default, and have to eliminate values of x that aren't in the domain to narrow it down. That means you need an x-value that sets the denominator to 0, so the result will be undefined (cannot divide by zero).
So, we set the denominator "1 - 2x" equal to 0:
1 - 2x = 0
1 = 2x
1/2 = x
The x-value 1/2 causes the function to be undefined, so we've found our domain: all real numbers EXCEPT x = 1/2.
God, the "helpers" here have gone downhill.
Actually Smart set the numerator equal to 0 which isnβt a restriction. 1 is the only restriction from g(x), since an undefined value cannot be plugged into f(x). G(x) is never -1 (asymptote) so there is no restriction there.
To find the simplified form for f [g(x)], we need to substitute g(x) into the function f(x) and simplify.
First, let's substitute g(x) into f(x):
f [g(x)] = f(x/(1-x))
Now, let's replace x in f(x) with x/(1-x):
f [g(x)] = (1 - (x/(1-x))) / (1 + (x/(1-x)))
To simplify this, we need to eliminate the fractions. To do that, we'll multiply both the numerator and the denominator by the common denominator (1-x):
f [g(x)] = ((1 - x) / (1-x) - (x/(1-x))) / ((1 + x) / (1 - x))
Now, simplify the numerator:
f [g(x)] = (1 - x - x) / (1 - x)
Combine like terms:
f [g(x)] = (1 - 2x) / (1 - x)
Thus, the simplified form for f [g(x)] is (1 - 2x) / (1 - x).
Now, let's determine the domain of this function. The domain represents all possible values of x for which the function is defined.
In this case, we need to consider any values of x that would make the denominator (1 - x) equal to zero. To find those values, we set 1 - x = 0 and solve for x:
1 - x = 0
x = 1
So, the function is undefined for x = 1. Therefore, the domain of f [g(x)] is all real numbers except x = 1.
f(x)=(1-x)/(1+x) and g(x)=x/(1-x)
f(g(x))
= f(x/(1-x) )
= [ 1 - x/(1-x)] / [1 + x/(1-x)]
multiply top and bottom by 1-x
= (1-x - x) / (1 - x + x)
= (1 - 2x)/1
= 1 - 2x
check: pick an unlikely x value and using my calculator
g(2.35) = -1.7407...
f(-1.7407) = -3.699999 or -3.7
using my answer of f(g(x)) = 1-2x
I got -3.7
(my answer is correct)
since in the original, x β Β±1
that restriction has to carry through, so
domain is any real number, except x = Β±1
[ 1 - (x/(1-x))] / [ 1 +(x/(1-x)) ]
[ 1-x - x]/ [ 1-x +x]
1-2x
if x = 1, g(x) is undefined