Problem 71 - A Graphical Approach to Precalculus with Limits

Question

Let the domain of \(f(x)\) be [-1,2] and the range be \([0,3] .\) Find the domain and range of the following. $$-f(x)$$

Step-by-Step Solution

Answer

Domain: \([-1, 2]\); Range: \([-3, 0]\).

Step 1: Analyze original function

The original function \(f(x)\) has a domain of \([-1, 2]\) and a range of \([0, 3]\). This means the function is defined for all \(x\) between -1 and 2, and it produces output values from 0 to 3.

Step 2: Determine the domain of \(-f(x)\)

The domain of \(-f(x)\) is the same as the domain of \(f(x)\) because negating a function does not change the set of inputs for which the function is defined. Therefore, the domain of \(-f(x)\) is \([-1, 2]\).

Step 3: Find the range of \(-f(x)\)

Negating a function multiplies all output values by -1. If the range of \(f(x)\) is \([0, 3]\), then the range of \(-f(x)\) will be the negation of this set, which is \([-3, 0]\). Each positive value becomes its negative counterpart.

Key Concepts

Understanding Negation of Function

When we talk about the negation of a function, we mean transforming the function by multiplying all of its output values by -1. This operation does not modify the domain, but it flips the sign of every element in the range.

Imagine a function as a machine that accepts inputs and produces outputs based on certain rules. By negating it, the machine still takes in the same inputs but now provides outputs that have opposite signs to those it originally produced.

- **Domain Remains the Same**: Since we're only flipping the values after computation, the input values (domain) remain unaffected.- **Range Flips**: Every positive output becomes negative, and every negative output becomes positive. For example, if a function's range is \[a, b\], the range of its negation would be \[-b, -a\].

This is useful in various applications, like inverting relationships or finding reflections across the x-axis in graphical representations.

Exploring Precalculus Problem

Precalculus problems often involve understanding and manipulating functions to prepare for the more complex calculus concepts. They form the foundation for understanding how different transformations affect functions.

The given problem is an example of such manipulation, where we have to apply the negation transformation to a function. Initially, we determine the domain and range of the function, \(f(x)\), before applying any transformations. This understanding helps us predict how transformations like negation affect a function.

Here are a few crucial steps in approaching such problems:

• **Identify Original Domain and Range**: Knowing where the function is defined (domain) and the possible outputs (range) is vital.
• **Apply the Transformation**: Whether it's negation or another transformation, understanding the rule of transformation is important.
• **Determine New Range/Behavior**: Accurately derive the effect of the transformation, such as flipping or stretching the range.

Precalculus invites us to analyze and question, setting the scene for calculus by exploring these foundational concepts.

Interval Notation Explained

Interval notation is a compact way of expressing the domain and range of a function. It is used to describe a set of numbers along a number line and provides a visual cue about which numbers are included in a domain or range.

Interval notation uses brackets and parentheses to specify which numbers are included in a range:

• **Brackets \[...\]**: Indicate that the endpoints are included in the interval. For example, \[-1, 2\] includes both -1 and 2.
• **Parentheses \( ... \)**: Indicate that the endpoints are not included. For example, \(0, 3\) includes all numbers between 0 and 3 but not 0 or 3.

This type of notation is vital for clearly stating the domain and range for functions, as it concisely presents information that can guide further mathematical operations and graphing.