Find the domain of the composite function f◦g.
Find the domain of the composite function f◦g.
F(x)= 3/(x-3); g(x)= 2/x
f◦g = f(g) = 3/(g-3)
= 3/((2/x)-3)
= 3/((2-3x)/x)
= (3x)/(2-3x)
It looks like the domain of f◦g is all reals except x=2/3, but we also have to exclude x=0, since g is not defined there.
Well, to find the domain of the composite function f◦g, we need to consider the restrictions of both functions f(x) and g(x).
First, let's look at g(x) = 2/x. The only restriction here is that x cannot be equal to 0, as division by zero is undefined. So we exclude 0 from the domain.
Next, we have f(x) = 3/(x-3). Here, we need to avoid values of x that would make the denominator, (x-3), equal to zero. So x cannot be equal to 3.
Now, for the composite function f◦g, we need to make sure that the values of x we choose for g(x) do not violate the restrictions of f(x).
Since g(x) has a restriction at x = 0, we need to throw out any values of x that make g(x) equal to 0.
Substituting g(x) into f(x), we have f(g(x)) = 3/((2/x) - 3).
Now, if we find the denominator of this expression, ((2/x) - 3), and set it equal to zero, we can determine any additional restrictions.
(2/x) - 3 = 0
2 - 3x = 0
-3x = -2
x = 2/3
So, x = 2/3 is an additional restriction for the composite function f◦g. Therefore, the domain of f◦g is all x-values except x = 0 and x = 2/3.
To find the domain of the composite function f◦g, we need to consider the restrictions on both the individual functions f and g.
The domain of g(x) is the set of all real numbers except x = 0, since division by zero is not defined. Therefore, the domain of g is (-∞, 0) U (0, ∞).
The domain of f(x) is all real numbers except x = 3, since f(x) is undefined when the denominator (x - 3) is equal to zero. So the domain of f is (-∞, 3) U (3, ∞).
Now, for the composite function f◦g, we need to determine the values of x that satisfy both the domain of g and the domain of f.
Since the domain of g is (-∞, 0) U (0, ∞), we need to find the values of x in this domain that also satisfy the domain of f, which is (-∞, 3) U (3, ∞).
Considering these conditions, the domain of the composite function f◦g is (-∞, 0) U (0, 3) U (3, ∞).
To find the domain of the composite function f◦g, we need to consider the domain of g(x) and the range of g(x) that is within the domain of f(x).
First, let's find the domain of g(x) by considering any possible restrictions. In this case, the only restriction is that the denominator cannot be zero, so x ≠ 0.
Next, we need to find the range of g(x) that is within the domain of f(x). Since g(x) is a rational function, the range is all real numbers except for the ones that make the denominator zero. Therefore, the range of g(x) that is within the domain of f(x) is x ≠ 0.
Finally, we combine the domain of g(x) (x ≠ 0) with the range of g(x) that is within the domain of f(x) (x ≠ 0). The common values between these two sets are x ≠ 0.
Therefore, the domain of the composite function f◦g is x ≠ 0.