Let ƒ(x) = √(x-2) and g(x) = x^2. Find each of the following, if possible. (g ○ ƒ)(x)

Key Concepts

Function Composition

Function composition involves combining two functions to create a new function. The notation (g ○ ƒ)(x) means to apply function ƒ first and then apply function g to the result. This process requires substituting the output of ƒ into g, which is essential for solving the problem.

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 the function ƒ(x) = √(x-2), the input must be greater than or equal to 2 to avoid taking the square root of a negative number. Understanding the domain is crucial when composing functions to ensure the resulting function is valid.

Evaluating Functions

Evaluating functions involves substituting a specific value into the function to find the corresponding output. In the context of function composition, after determining the domain, you will evaluate ƒ(x) first and then substitute that result into g(x). This step is vital for finding the final output of the composed function.