In Exercises 51–66, find a. (fog) (x) b. (go f) (x). f(x) = √x, g(x) = x − 1

Key Concepts

Function Composition

Function composition involves combining two functions to create a new function. If f(x) and g(x) are two functions, the composition (fog)(x) means applying g first and then applying f to the result, expressed as f(g(x)). Conversely, (go f)(x) means applying f first and then g, written as g(f(x)). Understanding this concept is crucial for solving the given problem.

Square Root Function

The square root function, denoted as f(x) = √x, is defined for non-negative values of x and returns the principal square root. This function is essential in the problem as it dictates the output of f when applied to the input from g. Recognizing the domain and range of the square root function is important for determining valid inputs and outputs during composition.

Linear Function

A linear function, such as g(x) = x - 1, represents a straight line when graphed and has a constant rate of change. In this case, g(x) shifts the input x down by 1. Understanding linear functions is vital for evaluating g(x) and subsequently applying f to the result, as it affects the overall behavior of the composed functions.