Hello! I seriously need help in functions! Thank you in advance!
Functions f and g are defined by f(x) = |x+1| , x is real, and g(x) = ln (x^2), x is real, x is not zero. Explain why fg exists and gf does not exist.
f(g(x)) exists if the range of g is a subset of the domain of f. The range of g is all real numbers. This is a subset of the domain of f. Therefore f(g(x)) exists.
Note: Any set is considered a subset of itself.
g(f(x)) exists if the range of f is a subset of the domain of g. The range of f is all real numbers greater than OR EQUAL TO zero. This is NOT a subset of the domain of g because the domain of g does NOT include zero. Therefore g(f(x)) does not exist.