Find fg and determine the domain for each function. f(x) = 2x + 3, g(x) = x − 1

Verified step by step guidance

1. Understand that the composition of functions fg means you substitute g(x) into f(x), so fg(x) = f(g(x)).
2. Write down the given functions: f(x) = 2x + 3 and g(x) = x - 1.
3. Substitute g(x) into f(x): replace every x in f(x) with g(x), so fg(x) = 2(g(x)) + 3.
4. Simplify the expression: fg(x) = 2(x - 1) + 3, then distribute and combine like terms.
5. Determine the domain of fg by considering the domain of g(x) first, then ensure that the output of g(x) fits into the domain of f(x). Since both f and g are polynomials, their domains are all real numbers.

Key Concepts

Function Composition

Function composition involves applying one function to the result of another, denoted as (fg)(x) = f(g(x)). This means you first evaluate g(x), then substitute that result into f. Understanding this process is essential to correctly find the composite function.

Domain of a Function

The domain of a function is the set of all input values (x) for which the function is defined. When composing functions, the domain of the composite function depends on the domain of the inner function and the domain restrictions of the outer function after substitution.

Linear Functions

Linear functions have the form f(x) = mx + b, where m and b are constants. They are defined for all real numbers, which simplifies domain considerations. Recognizing that both f and g are linear helps in quickly determining the domain of their composition.