Composition of Functions


Let f(x) = x − 3, g(x) = √x, h(x) = x³, and j(x) = 2x. Express each of the functions in Exercises 11 and 12 as a composition involving one or more of f, g, h, and j.


a. y = 2x − 3

Verified step by step guidance

1. Identify the functions provided: f(x) = x - 3, g(x) = √x, h(x) = x³, and j(x) = 2x.
2. Recognize that the function y = 2x - 3 can be expressed as a composition of the given functions.
3. Notice that j(x) = 2x is part of the expression y = 2x - 3, suggesting that j(x) is involved in the composition.
4. Observe that f(x) = x - 3 is also part of the expression y = 2x - 3, indicating that f(x) is involved in the composition.
5. Express y = 2x - 3 as a composition of j and f: y = f(j(x)).

Key Concepts

Composition of Functions

Composition of functions involves combining two or more functions to create a new function. If you have functions f(x) and g(x), the composition is denoted as (f ∘ g)(x) = f(g(x)). This means you apply g first and then apply f to the result. Understanding how to manipulate and combine functions is essential for solving problems that require expressing one function in terms of others.

Linear Functions

A linear function is a polynomial function of degree one, typically expressed in the form y = mx + b, where m is the slope and b is the y-intercept. In the context of the given problem, the function y = 2x - 3 is linear, indicating a constant rate of change. Recognizing the characteristics of linear functions helps in identifying how to express them using other functions.

Function Transformation

Function transformation refers to the changes made to a function's graph through operations such as shifting, stretching, or reflecting. In the case of y = 2x - 3, the function can be seen as a transformation of the basic linear function y = 2x, shifted down by 3 units. Understanding transformations is crucial for expressing functions in terms of others, as it allows for the identification of how one function can be derived from another.