a) given f(x)=1/(x^2-4) and g(x)=log(x), identify the steps you would take to determine the domain of gof(x). What is the domain of gof(x)?
b)Would the domain of fog(x) be the same? Explain. What is the domain of fog(x)?
check the related questions below.
oops my bad. Thank you!
a) To determine the domain of gof(x), we need to find the values of x that make the composition function gof(x) well-defined.
1. Start with the domain of g(x) (the inner function). In this case, g(x) = log(x).
The natural logarithm function, log(x), is defined only for positive values of x. Therefore, the domain of g(x) is x > 0.
2. Next, consider the domain of f(x) (the outer function). In this case, f(x) = 1/(x^2-4).
The expression x^2 - 4 is in the denominator, so we need to identify the values of x that make this expression non-zero. Solving the equation x^2 - 4 = 0, we get x = -2 and x = 2. Thus, these values become the excluded values for the domain of f(x).
3. Now, we combine the domains of g(x) and f(x) to find the domain of gof(x) (the composition function).
Since g(x) is defined for x > 0 and f(x) is defined for all real numbers except x = -2 and x = 2, the domain of gof(x) will be the intersection of these two domains: x > 0, excluding x = -2 and x = 2.
Therefore, the domain of gof(x) is (0, -2) U (-2, 0) U (0, 2) U (2, ∞).
b) The domain of fog(x) may not necessarily be the same as the domain of gof(x) because the order of composition matters.
The domain of fog(x) is determined by the domain of f(x), the outer function. In this case, f(x) = 1/(x^2-4).
Following the same steps as in part (a), we find that the domain of fog(x) is all real numbers except x = -2 and x = 2.
Therefore, the domain of fog(x) is (-∞, -2) U (-2, 2) U (2, ∞).