For f of x equals the quotient of the quantity 1 minus x and the quantity 1 plus x and g of x equals the quotient of the quantity x and the quantity 1 plus x, find the simplified form for f [g(x)] and state the domain.
why all the wordy mumbo jumbo? Ever heard of math symbols?
f(x) = (1-x)/(1+x)
g(x) = x/(1+x)
f(g) = (1-g)/(1+g)
= (1-(x/(1+x)))/(1+(x/(1+x)))
= 1/(2x+1)
the domain excludes x = -1, since g is not defined there. It also exludes x = -1/2, as shown by the final form of the fraction.
f(x)=(1-x)/(1+x)
g(x)=x/(1+x)
f(g(x))=(1-[x/(1+x)])/(1+[x/1+x])
f(g(x))=((1+x-x)/(1+x)/((1+x+x)/(1+x))
f(g(x))=(1/(1+x))*(1+x)/(1+2x)
f(g(x()=1/(1+2x)
the domain is all real numbers except any number that makes the denominator equal to 0
so, 1+2x=0, 2x=-1, x=-1/2
the domain is all real numbers except -1/2
To find the simplified form of f[g(x)], we need to substitute the expression for g(x) into f(x) and simplify.
Given that f(x) = (1 - x)/(1 + x) and g(x) = x/(1 + x), we can substitute g(x) into f(x) as follows:
f[g(x)] = f(x/(1 + x))
Next, we substitute the expression for g(x) into f(x):
f[g(x)] = [(1 - (x/(1 + x)))/(1 + (x/(1 + x))]
Now, let's simplify the expression:
f[g(x)] = [(1 - (x/(1 + x)))/((1 + (x/(1 + x))]
To simplify further, we need to find a common denominator for the fractions in the numerator:
f[g(x)] = [(1(1 + x) - x)/((1 + x) + (x/(1 + x))]
f[g(x)] = [(1 + x - x)/((1 + x) + (x/(1 + x))]
Canceling out the x terms in the numerator:
f[g(x)] = (1/(1 + x))
So, the simplified form of f[g(x)] is 1/(1 + x), and the domain is all real numbers except x = -1.
To find the simplified form for f[g(x)], we need to substitute g(x) into the expression for f(x) and simplify.
First, let's find g(x) by substituting x into the expression for g(x):
g(x) = x/(1 + x)
Next, substitute g(x) into the expression for f(x):
f[g(x)] = f[x/(1 + x)]
Now, let's substitute the expression for g(x) into f(x):
f[g(x)] = f[x/(1 + x)] = (1 - (x/(1 + x)))/(1 + (x/(1 + x)))
To simplify, we can simplify the numerator and denominator separately:
Numerator:
1 - (x/(1 + x)) = (1 + x - x)/(1 + x) = 1/(1 + x)
Denominator:
1 + (x/(1 + x)) = (1 + x + x)/(1 + x) = (1 + 2x)/(1 + x)
Putting it all together, the simplified form of f[g(x)] is:
f[g(x)] = 1/(1 + x)/(1 + 2x)/(1 + x)
Now, let's simplify further by multiplying the numerator and denominator by the reciprocal of the denominator:
f[g(x)] = (1/(1 + x)) * ((1 + x)/(1 + 2x))
Canceling out the common terms in the numerator and denominator, we get:
f[g(x)] = 1/(1 + 2x)
The simplified form of f[g(x)] is 1/(1 + 2x).
Now, let's consider the domain. The only restriction in the domain arises when the denominator becomes zero since division by zero is undefined. Therefore, we need to find the values of x that make the denominator equal to zero:
1 + 2x = 0
Solving for x gives:
2x = -1
x = -1/2
So, the domain of f[g(x)] is all real numbers except x = -1/2.