Find the quotient function f/g for f(x)=sqr(x+1) and g(x)= sqr( x-1).
My Answer:
sqr(x+1)/sqr( x-1)
sqr(x^2-1)/ (x-1)
However, I also have to state the restrictions to the domain and range, which I do not know how to do. Could someone please help me? Thanks.
your answer is correct.
for the restriction, the denominator cannot be zero, so x≠1
also the inside of the square root cannot be negative,
this will happen when -1 ≤ x ≤ +1
so the domain is x ≤ -1 OR x > 1
as x gets +large, the expression approaches 1
as x gets -large, the expression approaches -1
range: y >1 OR y < -1
To find the quotient function f/g for f(x) = sqrt(x+1) and g(x) = sqrt(x-1), we need to divide f(x) by g(x) by forming a single rational expression.
Step 1: Write the given functions:
f(x) = sqrt(x+1)
g(x) = sqrt(x-1)
Step 2: Formulate the quotient function:
f/g = f(x) / g(x)
We substitute the given functions into the quotient function:
f/g = sqrt(x+1) / sqrt(x-1)
Step 3: Rationalize the denominator:
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator.
f/g = (sqrt(x+1) * sqrt(x-1)) / (sqrt(x-1) * sqrt(x-1))
f/g = (sqrt(x+1) * sqrt(x-1)) / (x-1)
Step 4: Simplify the expression:
To simplify the expression, multiply the square roots together:
f/g = sqrt[(x+1)(x-1)] / (x-1)
f/g = sqrt[x^2-1] / (x-1)
So the quotient function f/g is sqrt(x^2-1) / (x-1).
Now, let's discuss the domain and range restrictions:
Domain restrictions:
1. The function f/g is defined as long as the denominator, which is (x-1), is not equal to zero. Therefore, x cannot be equal to 1. So, the domain of the quotient function f/g is all real numbers except x = 1.
Range restrictions:
The range of f/g depends on the range of f(x) and g(x). As both f(x) and g(x) are square root functions, their range is nonnegative real numbers or [0, ∞). Since dividing by a positive number preserves the nonnegative property, the range of the quotient function f/g is also [0, ∞).
Thus, the domain of the quotient function f/g is all real numbers except x = 1, and the range is [0, ∞).