Problem 16 - A Graphical Approach to College Algebra

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 \(|f(x)|\) is non-negative; it cannot include negative numbers like \(-1\).

Step 1: Understanding Absolute Values

The expression \(y = |f(x)|\) represents the absolute value of the function \(f(x)\). By definition, the absolute value of any real number is either positive or zero, never negative.

Step 2: Analysis of Range

Since \(|f(x)|\) can only produce non-negative outcomes, the range of \(y = |f(x)|\) must consist of zero and all positive numbers generated by \(f(x)\). Negative numbers, such as \(-1\), cannot be part of this range.

Step 3: Conclusion

Since \(|f(x)|\) always results in non-negative values, a negative number cannot be within the range of \(y = |f(x)|\). Therefore, \(-1\) cannot be included in the range.

Key Concepts

Range of a Function

When talking about the range of a function, it's crucial to understand that it consists of all possible values that the function can produce. If we consider a function like \( y = |f(x)| \), we are specifically looking at the values that result from the absolute value operation applied to the original function \( f(x) \).

The absolute value is a mathematical operation that transforms any real number into its non-negative counterpart. This means the range can only include numbers that are zero or positive. Negative numbers, like \(-1\), are not possible unless the original function could produce them without the absolute value restriction. Therefore, knowing the range helps us understand what outputs we can expect and guides us in solving equations or comparing functions.

In practical terms:

• The range of \( y = |f(x)| \) is always \([0, \infty)\), meaning it starts at zero and goes up to positive infinity.
• This happens because absolute values convert negative outputs of \( f(x) \) to positive ones, influencing the range.

Non-Negative Values

Absolute value functions are inherently designed to produce non-negative values. The concept behind an absolute value, denoted by the vertical bars \( | \, | \), is simply to strip any negative sign from a number, leaving it positive or zero.

In terms of function outputs, if \( f(x) \) results in a negative value, \( |f(x)| \) turns it into a positive, maintaining the magnitude but dropping the negative sign. This is why the range of \( y = |f(x)| \) never includes negatives.

Consider these points:

• \( |a| = a \) if \( a \geq 0 \)
• \( |a| = -a \) if \( a < 0 \)

From this, it's clear that any absolute value calculation cleans out negativity, ensuring the result is zero or positive.This is why the outputs always lie in the set of non-negative numbers.

Function Transformation

Function transformation refers to changes made to a function's equation that affect its graph's shape or position. When you work with absolute value functions like \( y = |f(x)| \), you're applying a transformation that impacts the original function \( f(x) \).

These transformations affect how the function graphically represents each \( x \)-value. For absolute value functions, this transformation is called reflection. It reflects any part of the graph of \( f(x) \) that is below the \( x \)-axis upward, making the global graph only occupy the top part of the plane.

Key points to understand include:

• Reflection across the x-axis occurs if any \( y \) values of \( f(x) \) are negative. The reflection operation makes these positive in \( |f(x)| \).
• It ensures that all parts of the graph lie above or on the \( x \)-axis after applying absolute value.

Understanding function transformation helps us predict and manipulate what the function graph will look like after changes are made. This makes it crucial for studying and simplifying complex equations.