Problem 38 - Calculus with Concepts in Calculus

Question

Suppose \(f\) is defined on \([a, b]\) and \(g(x)=f(x+c)\) for a fixed \(c\). What is the domain of \(g\) ?

Step-by-Step Solution

Answer

The domain of \(g(x)\) is \([a-c, b-c]\).

Step 1: Understanding the Problem

The function \(f\) is defined for values in the interval \([a, b]\), which means \(f(x)\) is valid for any \(x\) within this interval. The function \(g(x) = f(x+c)\) modifies \(f\) by adding a constant \(c\) to the input \(x\). Our task is to determine the domain of \(g\).

Step 2: Analyzing the Transformation

The transformation affects \(x\) by shifting it to \(x + c\). This means every input for \(g\) is tied to a shifted input for \(f\). The shifted input \(x + c\) must be within the original domain \([a, b]\) of \(f\).

Step 3: Setting the Domain Conditions

For \(g(x)\) to be defined and valid, \(x + c\) must satisfy the inequality \(a \leq x + c \leq b\). This ensures that \(f(x + c)\) is also within its domain.

Step 4: Solve for x

Re-arrange the inequalities to solve for \(x\). Start with \(a \leq x + c\), which transforms to \(x \geq a - c\). Similarly, the condition \(x + c \leq b\) gives \(x \leq b - c\).

Step 5: Formulating the New Domain

Combining both inequalities, we conclude that \(x\) must satisfy \(a - c \leq x \leq b - c\). Therefore, the domain of \(g(x)\) is the interval \([a-c, b-c]\).

Key Concepts

Function Transformation

When we talk about function transformation, we are referring to how functions change when they are altered in various ways. In this exercise, we encounter a specific type of transformation: translation. Here, the function \(g(x) = f(x+c)\) represents a horizontal shift. This means every input \(x\) for the function \(f(x)\) is adjusted by adding a constant \(c\).

This shift affects how the function is defined over its domain. If \(c\) is positive, the graph of \(f(x)\) shifts to the left by \(c\) units, whereas if \(c\) is negative, it shifts to the right. This kind of transformation doesn't alter the shape of the function. Instead, it only changes the position of the graph on the x-axis.

Understanding these transformations is crucial because they describe how the outputs are generated based on shifted inputs.

Inequalities

Inequalities are mathematical expressions indicating that one quantity is less than or greater than another. In this problem, they play a critical role in determining the new domain of the transformed function \(g(x)\).

Given the original domain \([a, b]\) of \(f\), our goal is to find the values of \(x\) such that when transformed by adding \(c\), these transformed values still lie within \([a, b]\). To ensure this, we use inequalities:

• \(x + c \geq a\)
• \(x + c \leq b\)

Solving these inequalities gives the range of \(x\) values that fulfill this condition, thus helping us establish the domain of \(g(x)\). Inequalities are vital in restricting domains to ensure we respect the original range where the function \(f(x)\) is defined.

Interval Notation

When expressing the domain of a function, interval notation is a concise way to describe a set of values. Interval notation uses brackets to show which numbers are included or excluded in a set. For instance, \([a-c, b-c]\) describes all real numbers between \(a-c\) and \(b-c\), including the endpoints.

In the provided exercise, the interval notation \([a-c, b-c]\) informs us how the domain has shifted because of the transformation \(x + c\). This notation is especially useful because it provides a clear and visual representation of the domain.

It's important to remember that square brackets \([, ]\) mean the numbers are included in the set, known as closed intervals. In contrast, parentheses \((, )\) would mean those numbers are excluded, indicating open intervals. Understanding this notation is crucial for interpreting the domain and range of functions easily and accurately.