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

Key Concepts

Function Composition

Function composition involves combining two functions to create a new function. For functions ƒ and g, the composition (ƒg)(x) means applying g first and then applying ƒ to the result of g. This is mathematically expressed as (ƒg)(x) = ƒ(g(x)). Understanding this concept is crucial for solving the problem as it dictates the order of operations when evaluating the combined function.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For linear functions like ƒ(x) = 3x + 4 and g(x) = 2x - 7, the domain is typically all real numbers unless specified otherwise. Identifying the domain is essential for understanding the behavior of the composed function and ensuring that all inputs yield valid outputs.

Linear Functions

Linear functions are polynomial functions of degree one, represented in the form f(x) = mx + b, where m is the slope and b is the y-intercept. In this case, ƒ(x) = 3x + 4 and g(x) = 2x - 7 are both linear functions. Recognizing their properties, such as constant rates of change and straight-line graphs, is important for analyzing their composition and understanding their domains.