For each pair of functions below, find (a) h(x)=(f ∘g)(x) and (b) H(x)=(g ∘f)(x), and (c) determ

step 1 Okay, this problem starts us with two functions. We have f of x is equal to the absolute value o

For each pair of functions below, find (a) h(x)=(f ∘g)(x)...

Key Concepts

Function Composition

Function composition is the process of applying one function to the result of another. When you compose two functions, such as (f ? g)(x), you first evaluate g(x) and then apply f to that result. This concept is fundamental in mathematics because it allows for building more complex functions from simpler ones and requires careful attention to the order in which functions are applied.

Domain of Composite Functions

The domain of a composite function is determined by the domains of both the inner and outer functions. Specifically, for (f ? g)(x), the input x must be within the domain of g, and the output of g(x) must lie within the domain of f. Analyzing and determining these constraints is crucial to accurately define the domain of the composite function.

Absolute Value Function

An absolute value function returns the non-negative magnitude of a real number regardless of its sign. In many cases, such functions are defined for all real numbers, which simplifies determining their domain when they are part of a composite function. Their piecewise linear nature is also important when performing algebraic manipulations in compositions.

Linear Functions

Linear functions have the form ax + b and are characterized by a constant rate of change, meaning they are continuous and defined for all real numbers. Their simplicity makes them a common choice for composing with other functions because they do not typically introduce additional domain restrictions.

Transcript

Please give Ace some feedback