Given functions f and g, (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=-6x+9, g(x)=5x+7

Key Concepts

Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. In this case, (g∘ƒ)(x) means applying function f first and then applying function g to the result. Understanding how to correctly perform this operation is crucial for evaluating the composite 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. When composing functions, the domain of the composite function (g∘ƒ)(x) is determined by the domain of f and the values that f can take that are also in the domain of g. This ensures that the composition is valid.

Linear Functions

Linear functions are polynomial functions of degree one, typically expressed in the form f(x) = mx + b, where m is the slope and b is the y-intercept. The functions given, f(x) = -6x + 9 and g(x) = 5x + 7, are both linear, which means their graphs are straight lines. Understanding their properties helps in analyzing their compositions and behaviors.