Exploring Function Operations: Addition, Division, and Composition
As an excellent high school mathematics teacher, I'm excited to guide you through the exploration of various operations on functions. In this article, we'll dive into the concepts of function addition, division, and composition, and learn how to find the resulting functions and their respective domains.
a. f + g and its domain
Let's start by considering the functions f(x) = x + 5 and g(x) = (x^2) - 3. To find the function f + g, we simply add the corresponding function values:
(f + g)(x) = f(x) + g(x)
(f + g)(x) = (x + 5) + ((x^2) - 3)
(f + g)(x) = x + 5 + x^2 - 3
(f + g)(x) = x^2 + x + 2
The domain of the resulting function f + g is the intersection of the domains of f(x) and g(x). Since both f(x) and g(x) are defined for all real numbers, the domain of f + g is also the set of all real numbers, denoted as R.
b. g/f and its domain
Next, let's explore the division of the functions g(x) and f(x). The resulting function is denoted as g/f:
(g/f)(x) = g(x) / f(x)
(g/f)(x) = ((x^2) - 3) / (x + 5)
The domain of the function g/f is the set of all real numbers x such that f(x) ≠ 0. In this case, f(x) = x + 5, which is never equal to 0 for any real number x. Therefore, the domain of g/f is also the set of all real numbers, R.
c. f(g(0))
To find f(g(0)), we first need to evaluate g(0) and then substitute the result into f(x).
g(0) = (0^2) - 3 = -3
f(g(0)) = f(-3) = -3 + 5 = 2
Therefore, f(g(0)) = 2.
d. g(f(x))
The composition of the functions g(x) and f(x) is denoted as g(f(x)). To find g(f(x)), we substitute f(x) into g(x):
g(f(x)) = g(x + 5)
g(f(x)) = ((x + 5)^2) - 3
g(f(x)) = x^2 + 10x + 25 - 3
g(f(x)) = x^2 + 10x + 22
The domain of the resulting function g(f(x)) is the set of all real numbers x for which f(x) is defined, which in this case is the set of all real numbers, R.
I hope this detailed explanation has helped you understand the various operations on functions and their corresponding domains. If you have any further questions, feel free to ask!