Problem 59 - Algebra and Trigonometry

Question

Draw the graph of \(f\) and use it to determine whether the function is one-to- one. \(f(x)=|x|-|x-6|\)

Step-by-Step Solution

Answer

The function is not one-to-one.

Step 1: Understand Absolute Function Properties

The absolute value function is defined as \( |z| = z \) if \( z \geq 0 \), and \( |z| = -z \) if \( z < 0 \). We will use this property to determine cases for \( f(x) = |x| - |x-6| \).

Step 2: Analyze the Function for Different Intervals

Since the function involves two absolute value expressions, identify the critical points where the expression inside the absolute values change sign. These points are \( x = 0 \) and \( x = 6 \). Evaluate the behavior of \( f(x) \) over three intervals: \( (-\infty, 0) \), \( [0, 6] \), and \( (6, \infty) \).

Step 3: Calculate \(f(x)\) for \(-\infty < x < 0\)

In this interval, \( x < 0 \) leads to \( |x| = -x \), and \( x - 6 < 0 \) leads to \( |x-6| = -(x-6) = -x + 6 \). Thus, \( f(x) = -x - (-x + 6) = 6 \).

Step 4: Calculate \(f(x)\) for \(0 \leq x < 6\)

In this interval, \( x \geq 0 \) gives \( |x| = x \), and \( x - 6 < 0 \) gives \( |x-6| = -(x-6) = -x + 6 \). So, \( f(x) = x - (-x + 6) = 2x - 6 \).

Step 5: Calculate \(f(x)\) for \(x \geq 6\)

In this interval, both expressions are non-negative, \( x \geq 0 \) and \( x - 6 \geq 0 \), so \( |x| = x \) and \( |x-6| = x - 6 \). Therefore, \( f(x) = x - (x - 6) = 6 \).

Step 6: Sketch the Graph of \(f(x)\) by Piecewise Functions

Based on previous calculations, plot the graph: \( f(x) = 6 \) for \( x < 0 \), \( f(x) = 2x-6 \) for \( 0 \leq x < 6 \), and \( f(x) = 6 \) for \( x \geq 6 \). The graph is a horizontal line \( y = 6 \) for \( x < 0 \) and \( x \geq 6 \), and a line with slope 2 for \( 0 \leq x < 6 \).

Step 7: Determine If the Function is One-to-One

A function is one-to-one if no horizontal line intersects the graph at more than one point. Since \( f(x) = 6 \) for both \( x < 0 \) and \( x \geq 6 \), a horizontal line at \( y = 6 \) intersects the graph in more than one region, showing \( f(x) \) is not one-to-one.

Key Concepts

Absolute Value

The absolute value function is an interesting one! It's based on distance rather than direction. Essentially, the absolute value of a number is how far it is from zero on the number line, regardless of direction. So, if we are dealing with a number greater than or equal to zero, the absolute value of that number is simply the number itself. That's where the rule \( |z| = z \) for \( z \geq 0 \) comes in.

• If \( z = 5 \), then \( |z| = 5 \).
• If \( z = 0 \), then \( |z| = 0 \) because zero itself is its absolute value.
• If \( z = -3 \), the absolute value changes to 3 as we consider only distance, giving us \( |z| = -(-3) = 3 \).

Now, applying this to any expressions involving absolute values, like \( f(x) = |x| - |x-6| \), we need to determine each absolute value separately, based on the conditions where they switch from positive to negative. Points like \( x = 0 \) and \( x = 6 \) become crucial because they change the underlying expression inside the absolute value signs.

Function Graphing

Graphing functions like \( f(x) = |x| - |x-6| \) helps us visualize behavior over different ranges. For this function, after understanding the absolute values, we break our graph into sections and examine each separately.

• **For \( x < 0 \):** We find that \( f(x) = 6 \). This is a horizontal line, simple yet important. Graph it straight across the vertical axis at \( y = 6 \).
• **For \( 0 \leq x < 6 \):** The function \( f(x) = 2x-6 \) is linear with a slope of 2. It rises quickly, starting at \( y = -6 \) (as found by substituting \( x = 0 \)).
• **For \( x \geq 6 \):** Again, we see \( f(x) = 6 \), meaning another horizontal stretch along \( y = 6 \).

Sketch these segments together to see the overall graph. You’ll notice it zigzags from horizontal to angled and back again. Each piece tells a story, and connecting them reveals the story of \( f(x) \).

Piecewise Functions

Piecewise functions like \( f(x) = |x| - |x-6| \) can be thought of as having multiple personalities. They're made up of different expressions in different parts of their domain. Let's break down how this works:

• In \( (-\infty, 0) \), \( f(x) = 6 \) is a constant function. It doesn't rise or fall; it stays put at 6.
• Between \( [0, 6) \), \( f(x) = 2x - 6 \). This piece is a line that behaves differently depending on \( x \).
• For \( x \geq 6 \), \( f(x) \) returns to the constant value 6.

When you graph these, you switch between rules depending on the interval you're working in. Piecewise functions add versatility in modeling real-world situations or abstract exercises where a single rule just won't do.