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

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Identify the functions provided: f(x) = x - 3, g(x) = √x, h(x) = x³, and j(x) = 2x.

Recognize that the function y = 2x - 3 can be expressed as a composition of the given functions.

Notice that j(x) = 2x is part of the expression y = 2x - 3, suggesting that j(x) is involved in the composition.

Observe that f(x) = x - 3 is also part of the expression y = 2x - 3, indicating that f(x) is involved in the composition.

Express y = 2x - 3 as a composition of j and f: y = f(j(x)).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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