Let f (x)= (x^2 + 3x - 4) and g(x)= (x + 4)
a. Find f*g and state the domain
b. Find f/g and state the domain
the domain of all polynomials is (-∞,+∞)
for rational functions, it is the same, except where the denominator is zero.
f*g = (x^2+3x-4)(x+4)
f/g = (x^2+3x-4)/(x+4)
To find the composite function of f * g, we substitute g(x) into f(x).
a. Composite Function f * g:
f * g = f(g(x)) = f(x + 4)
To find f(x + 4), we substitute (x + 4) into f(x).
f(x + 4) = [(x + 4)^2 + 3(x + 4) - 4]
Expanding the equation, we get:
f(x + 4) = [x^2 + 8x + 16 + 3x + 12 - 4]
Combining like terms, we have:
f(x + 4) = x^2 + 11x + 24
The domain of the function f * g is the same as the domain of f(x) and g(x), which is all real numbers.
Now, let's move on to the next question.
b. Division of the Functions f/g:
To find f/g, we divide f(x) by g(x).
f/g = f(x) / g(x) = (x^2 + 3x - 4) / (x + 4)
However, we need to be careful when we divide by zero, as it is undefined. We must determine the values that make the denominator equal to zero and exclude them from the domain.
Setting the denominator equal to zero and solving for x:
x + 4 = 0
x = -4
So, the value x = -4 is not in the domain because it would make the denominator zero.
Therefore, the domain of f/g is all real numbers except x = -4.
To find the product f*g of the functions f(x) and g(x), we need to multiply the two functions together.
a. Finding f*g:
We substitute g(x) = (x + 4) into f(x) = (x^2 + 3x - 4), and simplify the expression:
f*g = f(x) * g(x)
= (x^2 + 3x - 4) * (x + 4)
= x^3 + 4x^2 + 3x^2 + 12x - 4x - 16
= x^3 + 7x^2 + 8x - 16
To find the domain of f*g, we need to consider the restrictions on the values of x that make the expression defined. In this case, since both f(x) and g(x) are polynomial functions, they are defined for all real numbers. Therefore, the domain of f*g is all real numbers.
b. Finding f/g:
To find the quotient f/g of the functions f(x) and g(x), we divide f(x) by g(x).
f/g = f(x) / g(x)
= (x^2 + 3x - 4) / (x + 4)
However, before finding the domain, we need to check if there are any values of x for which g(x) equals zero. If g(x) = 0, then the function g(x) is undefined at those values. In this case, we find the value of x that makes g(x) = 0:
x + 4 = 0
x = -4
Therefore, x = -4 is the value that makes g(x) equal to zero. We must exclude this value from the domain when calculating f/g.
Now, to find the domain of f/g, we need to ensure that the function is defined for all other values of x except x = -4. So, the domain of f/g is all real numbers except x = -4.