For the pair of functions defined, find (ƒg)(x).Give the domain of each. See Example 2.
ƒ(x)=3x+4, g(x)=2x-7

Verified step by step guidance

1. Understand that (ƒg)(x) means the composition of functions, which is ƒ(g(x)). This means you will substitute the entire function g(x) into the function ƒ(x).
2. Write down the given functions: ƒ(x) = 3x + 4 and g(x) = 2x - 7.
3. Substitute g(x) into ƒ(x): replace every x in ƒ(x) with g(x), so (ƒg)(x) = 3(2x - 7) + 4.
4. Simplify the expression by distributing and combining like terms: multiply 3 by each term inside the parentheses and then add 4.
5. Determine the domain of each function: since both ƒ(x) and g(x) are linear functions, their domains are all real numbers, but confirm this by considering any restrictions from the composition.

Key Concepts

Function Composition

Function composition involves applying one function to the result of another, denoted as (ƒg)(x) = ƒ(g(x)). This means you first evaluate g(x), then substitute that result into ƒ(x). 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 domains of both functions and the outputs of the inner function.

Linear Functions

Linear functions have the form f(x) = mx + b, where m and b are constants. They are defined for all real numbers, making their domain all real numbers. Recognizing this helps simplify finding domains in compositions involving linear functions.