What is the domain of
(fg)(x)
f(x)=x2–4
g(x)=3x2
answer choices are
0
all real numbers except -2 and 2
all real numbers except 0
-2 and 2
all real numbers
does anybody know the answer
I will read that as
(f (g(x) )
= f(3x^2)
= (3x^2)^2 - 4
= 9x^4 - 4
Clearly the domain is the set of reals
We could have seen that without actually doing any work
Since no division took place, (worry about dividing by zero),
and no operations such as square roots were found, it was
obvious that we would get results no matter what x's we used.
or, if (fg)(x) = f(x)*g(x) the same logic applies. It's still just a polynomial, and the domain is all real numbers.
To determine the domain of (fg)(x), we need to find the values of x for which the composition (fg)(x) is defined.
First, let's break down what (fg)(x) means. It is the composition of two functions, f(x) and g(x), where f(x) is x² – 4 and g(x) is 3x².
To find the composition (fg)(x), we substitute g(x) into f(x), like this:
(fg)(x) = f(g(x))
Substituting g(x) into f(x), we get:
(fg)(x) = f(3x²) = (3x²)² – 4
Simplifying further:
(fg)(x) = 9x⁴ – 4
Now, to determine the domain of (fg)(x), we need to identify any values of x that would cause a problem, such as division by zero or taking the square root of a negative number.
Since (fg)(x) is a polynomial function, there are no such restrictions. Thus, the domain of (fg)(x) is all real numbers, which is the last option among the answer choices given: "all real numbers."