f(x)=x^2+x-12
g(x)=x-3
Solve (f/g)(x) and state the domain
wondering if you mean f(g(x))
I think she means
(x^2 + x -12)/(x-3)
= (x+4)(x-3)/(x-3)
= x + 4 , x ≠3
(f/g)(x) = x+4 , x ≠ 3
domain: any real number, x≠3
To solve (f/g)(x), we need to divide the function f(x) by g(x).
First, let's divide each term of f(x) by g(x):
f(x)/g(x) = (x^2 + x - 12) / (x - 3)
To simplify this expression, let's factor the numerator and cancel out any common factors with the denominator.
Factoring the numerator:
(x^2 + x - 12) = (x + 4)(x - 3)
Now we can rewrite the expression:
f(x)/g(x) = [(x + 4)(x - 3)] / (x - 3)
Next, we can cancel out the common factor of (x - 3):
f(x)/g(x) = (x + 4)
The simplified form of (f/g)(x) is (x + 4).
Now let's determine the domain of the function (f/g)(x).
The domain represents all the values that x can take without causing division by zero or any other undefined operation.
In this case, the denominator g(x) = x - 3. To avoid division by zero, we need to exclude any value of x that makes the denominator equal to zero.
Thus, the domain of (f/g)(x) is all real numbers except x = 3.
Therefore, the domain is (-∞, 3) U (3, +∞).