Problem 72 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function and determine whether it is even, odd, or neither. $$f(x)=-|x-5|$$

Step-by-Step Solution

Answer

The function \(f(x)=-|x-5|\) is neither even nor odd.

Step 1: Graph the Function

The function can be graphed using a graphing utility such as an online graphing calculator. The vertical reflection is represented as '-' in the function, and the horizontal shift is represented as '-5' within the absolute value bars.

Step 2: Determine Evenness/Oddness

Check if the graph is symmetric around the y-axis (for even function) or origin (for odd function). If it is not symmetric about either the y-axis or origin, then it's neither even nor odd.

Step 3: Validate by Plugging Values

To ensure correctness, substitute \(x\) with \(-x\) and check if the function satisfies the conditions of being even or odd or neither. Calculate both \(f(x)\) and \(f(-x)\) and compare them.

Key Concepts

Even and Odd Functions

Understanding whether a function is even or odd involves checking its symmetry properties. For a function to be **even**, it must be symmetric around the y-axis. This means

• If you replace every instance of x with -x in the function, you'll get the same result.
• Mathematically, that's saying: if \( f(x) = f(-x) \), it's even.

On the other hand, if a function is **odd**, it will have symmetry around the origin. This means

• Replacing x with -x should result in the opposite sign of the original function.
• So, for odd functions: if \( f(-x) = -f(x) \), it's odd.

If a function does not satisfy either condition, it's considered neither even nor odd. With \( f(x) = -|x-5| \), substituting \( x \) with \( -x \) will show that this function doesn’t meet the criteria for either classification.

Graphing Utilities

Graphing utilities are tools, like online calculators, that help visualize mathematical functions. These tools make analyzing functions much easier. In examining the graph of \( f(x) = -|x-5| \), a graphing utility can quickly show us the shape and key features of the graph.

Key steps in using graphing utilities include:

• Entering the function: Type the function exactly as it is into the calculator or software.
• Observing symmetry: Look for visual symmetry about the y-axis or origin.
• Interpreting shifts: The term inside the absolute value \(-5\) shifts the graph horizontally.

These visual tools save time and improve understanding by quickly showing changes and effects represented in the function. As with \( f(x) = -|x-5| \), the graphing utility helps confirm it has a vertical flip and a horizontal shift.

Absolute Value Functions

Absolute value functions are unique because they always yield non-negative results, but they can transform in interesting ways. The basic form \( |x| \) produces a V-shaped graph.

With \( f(x) = -|x-5| \), there are significant changes to the usual appearance:

• The negative sign in front flips the V-shape upside down, making it an inverted V.
• The term \(-5\) inside shifts the graph 5 units to the right on the x-axis.

These elements together generate the downward-facing V, starting at \( x = 5 \). Absolute value functions can appear in various applied contexts and learning to transform them is a valuable skill. This function illustrates how algebraic modifications impact graph shapes clearly.