f(x)= x^2-16 and g(x)= x+. find f/g and it’s domain
f/g = f(x)/g(x) = (x^2-16)/(x+?)
I expect you meant g(x) = x+4, since that would make
f/g = (x^2-16)/(x+4) = x-4
That is true for all real numbers except x = -4, as that would make f/g = 0/0, which is undefined.
To find the quotient of two functions, we need to divide each term of the numerator function by the denominator function. In this case, we want to find (f/g)(x).
Let's start by substituting the expressions for f(x) and g(x) into the quotient:
(f/g)(x) = (x^2 - 16)/(x+)
Notice that you did not provide an expression for g(x). Please provide the correct expression for g(x) so we can continue solving the problem.
To find f/g, we need to divide the function f(x) by g(x).
f(x) = x^2 - 16
g(x) = x
Dividing f(x) by g(x), or f/g, means dividing each term of f(x) by g(x).
f/g = (x^2 - 16) / x
Now, let's simplify the expression:
Using polynomial long division, we divide x^2 - 16 by x:
1
_____________________
x | x^2 - 16
x(x - 0)
-----------
x | x^2 - 16
x^2 - 0
------------------
-16
The remainder is -16.
Therefore, the expression f(x)/g(x) simplifies to:
f/g = x - 0 - (16 / x)
f/g = x - 16/x
The domain of f/g is determined by the denominator, which is x.
Since division by zero is undefined, we need to exclude any x values that would make the denominator equal to zero. Thus, the domain of f/g is all real numbers except x = 0.
In interval notation, the domain of f/g is (-∞, 0) U (0, ∞).