Exploring Function Operations and Composition

As a high school mathematics teacher, I'm excited to guide you through the fascinating world of function operations and composition. In this article, we'll dive into the following concepts:

a. Finding the sum of two functions, f(x) = x + 5 and g(x) = (x^2) - 3, and determining its domain. b. Evaluating the quotient of the two functions, g/f, and identifying its domain. c. Calculating the value of f(g(0)). d. Determining the function g(f(x)).

Let's get started!

a. Finding f + g and its domain

To find the sum of the two functions, f(x) = x + 5 and g(x) = (x^2) - 3, 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 sum function, (f + g)(x), 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)(x) is also the set of all real numbers, denoted as ℝ.

b. Evaluating g/f and its domain

To find the quotient of the two functions, g/f, we divide the function values:

(g/f)(x) = g(x) / f(x) (g/f)(x) = ((x^2) - 3) / (x + 5)

The domain of (g/f)(x) is the set of all real numbers x such that f(x) ≠ 0. In this case, f(x) = x + 5, and x + 5 = 0 has a single solution, x = -5.

Therefore, the domain of (g/f)(x) is the set of all real numbers x except for x = -5, which is denoted as ℝ \ {-5}.

c. Calculating f(g(0))

To find the value of f(g(0)), we first need to evaluate g(0) and then substitute it into f(x).

g(0) = (0^2) - 3 = -3 f(g(0)) = f(-3) = -3 + 5 = 2

Therefore, f(g(0)) = 2.

d. Determining g(f(x))

To find the function g(f(x)), we need to substitute f(x) into g(x).

f(x) = x + 5 g(f(x)) = g(x + 5) = ((x + 5)^2) - 3

Therefore, g(f(x)) = (x + 5)^2 - 3.

I hope this article has helped you understand the various operations and compositions of functions. If you have any further questions, feel free to reach out to me. Happy learning!