Problem 16 - A Graphical Approach to Precalculus with Limits

Question

Give a short answer to each question. Why can't the range of \(y=|f(x)|\) include \(-1,\) for any function \(f ?\)

Step-by-Step Solution

Answer

The range of \(|f(x)|\) can't include \\-1\\ because absolute values are always non-negative.

Step 1: Understand the Absolute Value Function

The absolute value of a function, denoted as \(|f(x)|\), represents the distance of the function value from zero on the number line, which is always non-negative. This means it can only be zero or positive.

Step 2: Identify the Possible Range

Since the absolute value is non-negative, the range of \(|f(x)|\) includes all real numbers that are greater than or equal to zero. Specifically, it starts from 0 and goes up to positive infinity.

Step 3: Explain Why Negative Values Are Excluded

Given that the absolute value only yields non-negative results, negative numbers cannot appear in the range of \(|f(x)|\). Thus, a negative number like \-1\ is impossible to achieve with any function's absolute value.

Key Concepts

Range of Functions

The range of a function refers to the set of all possible output values it can produce. When we're dealing with functions in mathematics, understanding the range is crucial. It provides insight into what outputs are feasible when you input different values into the function.

In the case of the absolute value function, \(|f(x)|\), the range is notably distinct. Since absolute value measures the distance from zero, the range includes only non-negative values. This means:

• The lowest value is 0, because that's the smallest distance anything can be from zero.
• There is no maximum value, so technically, the range extends to infinity as the function's input value becomes increasingly positive or negative.

The range of \(|f(x)|\) is therefore \([0, \infty)\), meaning it covers all real numbers starting from zero going upwards. Negative numbers, like \-1\, are never in the range of \(|f(x)|\), since you can't have a negative distance from zero.

Non-Negative Values

The concept of non-negative values is essential when discussing absolute value functions. An absolute value function \(f(x)\) evaluates to the distance a number is from zero. Since distance itself can never be negative, absolute values are always non-negative.

A number is non-negative if it is either positive or zero but never negative. This applies directly to absolute value functions for any input, guaranteeing that \(|f(x)|\) results in:

• Zero, if the input value makes the original function value zero.
• A positive number, if the original function value is non-zero, whether positive or negative.

Understanding this principle assures that every output of \(|f(x)|\) is a valid non-negative number in the real number line.

Function Properties

Function properties refer to the characteristics that determine a function's behavior and output values, including aspects like continuity, domain, and range.

For absolute value functions, crucial properties include:

• Non-negativity: As previously discussed, the absolute value function's outputs are always zero or positive. This is a core property that shapes what the function looks like graphically (a V-shaped graph with the tip at the origin).
• Continuity: Absolute value functions are continuous, which means their graphs have no breaks or gaps.
• Symmetry: These functions are symmetric with respect to the y-axis. This means the graph appears mirrored on either side of the y-axis.
• Piecewise Definition: The absolute value function can be defined with a piecewise function, such as \(|x| = x\) if \(x \geq 0\) and \(-x\) if \(x < 0\).

Such properties are integral for solving absolute value problems and understanding their implications in real-world contexts.