Given f(x)=1/x^2 -4 and g(x)=log(x) identify the steps you would take to determine the domain of (g ∘ f)(x). What is the domain of (g ∘ f)(x)?
I assume you mean
f(x) = 1/(x^2-4)
If so, then
(g ∘ f)(x) = g(f) = log(f) = log(1/(x^2-4)) = -log(x^2-4)
so, the domain of g is where f > 0; that is, where |x| > 2
(f ∘ g)(x) = f(g) = 1/((logx)^2-4)
So now we need x>0 and logx≠2
To determine the domain of the composition function (g ∘ f)(x), we need to consider the domains of both functions f(x) and g(x) and determine which values of x are allowed for their composition.
Step 1: Find the domain of f(x):
The expression f(x) = 1/x^2 - 4 involves a fraction and a square, so let's examine the restrictions:
1. The denominator x^2 cannot be zero, as division by zero is undefined. So we need to ensure x^2 ≠ 0.
2. Additionally, the function also has a subtraction of 4, which poses no specific restrictions.
Hence, the domain of f(x) is all real numbers except x = 0. We can write it as: \[ \text{Domain of } f(x): (-\infty, 0) \cup (0, \infty) \]
Step 2: Find the domain of g(x):
The function g(x) = log(x) involves a logarithm, which is only defined for positive values. Thus, the argument x of the logarithm must be greater than zero: x > 0.
Hence, the domain of g(x) is all positive real numbers. We can write it as: \[ \text{Domain of } g(x): (0, \infty) \]
Step 3: Determine the domain of (g ∘ f)(x):
To find the domain of (g ∘ f)(x), we need to ensure that the values obtained from f(x) fall within the domain of g(x).
By performing the composition (g ∘ f)(x) = g(f(x)), we substitute f(x) into g(x) as follows: g(f(x)) = log(1/x^2 - 4)
To find the domain of (g ∘ f)(x), we have to take into account two restrictions:
1. The argument of the logarithm, 1/x^2 - 4, must be greater than zero: 1/x^2 - 4 > 0. This step is necessary because we are applying a logarithm to it.
2. The x-values must also fall within the domain of f(x), excluding x = 0.
To find the first restriction, we can solve the inequality:
1/x^2 - 4 > 0
Adding 4 to both sides: 1/x^2 > 4
Reciprocal of both sides (since 1/(1/x^2) = x^2): x^2 < 1/4
Taking the square root of both sides, and since we are only interested in positive values, we neglect the negative root: x < 1/2
To summarize, the domain of (g ∘ f)(x) is where the values of x satisfy both restrictions:
\[ \text{Domain of } (g \circ f)(x): (0, \frac{1}{2}) \]