Problem 69 - 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 as it outputs 6 for many values of \(x\).

Step 1: Understand the Function

The function given is \(f(x) = |x| - |x-6|\). This is a piecewise function consisting of absolute values, which means it behaves differently based on the domain intervals of \(x\).

Step 2: Analyze Domains and Break into Pieces

The function is influenced by the values within the absolute value brackets: \(|x|\) and \(|x-6|\). The key points are where these expressions change: at \(x = 0\) and \(x = 6\). We need to consider intervals: \(x < 0\), \(0 \leq x < 6\), and \(x \geq 6\).

Step 3: Calculate Function for Each Interval

1. For \(x < 0\), \(|x| = -x\) and \(|x-6| = -(x-6) = -x + 6\). Therefore, \(f(x) = (-x) - (-x + 6) = 6\).2. For \(0 \leq x < 6\), \(|x| = x\) and \(|x-6| = -(x-6) = -x + 6\). Therefore, \(f(x) = x - (-x + 6) = 2x - 6\).3. For \(x \geq 6\), \(|x| = x\) and \(|x-6| = x-6\). Therefore, \(f(x) = x - (x-6) = 6\).

Step 4: Graph the Function for Each Interval

- For \(x < 0\), the function is a horizontal line: \(f(x) = 6\).- For \(0 \leq x < 6\), it describes a linear function: \(f(x) = 2x - 6\), starting at point (0, -6) and increasing to (6, 6).- For \(x \geq 6\), it returns to a horizontal line: \(f(x) = 6\).Combine these to graph the entire piecewise function.

Step 5: Check If Function is One-to-One

For a function to be one-to-one, each value of \(x\) must correspond to a unique value of \(f(x)\), meaning no horizontal line should intersect the graph more than once.- For \(x < 0\) and \(x \geq 6\), \(f(x) = 6\) repeatedly, intersecting any horizontal line at \(6\) indicating lack of one-to-one.- For \(0 \leq x < 6\), \(f(x) = 2x - 6\) is a line with a unique output for each input, but the entire function needs to satisfy the condition, not just this part.

Key Concepts

Absolute Value Function

An absolute value function is defined as a piecewise function that takes a real number and returns its non-negative magnitude without regard to its sign. This means that whether the input number is positive or negative, the output will always be positive or zero. The notation for absolute value is given as \(|x|\), where for any real number \(|x|\), \(|x| = x\) if \(x \geq 0\) and \(|x| = -x\) if \(x < 0\). This characteristic makes absolute value functions especially useful in dealing with cases where only the size of a number matters, not its sign.
In the given function \(f(x) = |x| - |x-6|\), the absolute values create different behaviors over various intervals of x:

• For \(x < 0\), both expressions \(|x|\) and \(|x-6|\) become negative, simplifying \(f(x)\) into a constant value.
• For \(0 \leq x < 6\), \(|x|\) turns positive while \(|x-6|\) remains negative, resulting in a linear function.
• For \(x \geq 6\), both expressions become positive, ultimately providing another constant value.

This analysis highlights how absolute value functions can change the structure of a function depending on its domain, influencing its properties significantly.

One-to-One Function

A one-to-one function is crucial when considering whether a function can be inverted. It must ensure that every input corresponds to one and only one output, with no overlap in the outputs for different inputs. This requirement means that no horizontal line should intersect the graph of the function more than once, ensuring that the function passes the horizontal line test.
In the case of the function \(f(x) = |x| - |x-6|\), the analysis of whether it is one-to-one is a key objective:

• For \(x < 0\) and \(x \geq 6\), \(f(x)\) constantly equals 6, leading these segments of the graph to fail the horizontal line test because any horizontal line at \(y = 6\) intersects infinitely.
• For \(0 \leq x < 6\), the linear segment \(f(x) = 2x - 6\) itself would pass the test, yet it's not enough since one-to-one is a property the entire function must hold.

Thus, when considering the entire function, \(f(x)\) is not one-to-one due to constant values at endpoints making this feature unmet, stressing the necessity of examining all segments of piecewise functions when assessing one-to-one properties.

Graphing Functions

Graphing functions, particularly piecewise functions, can seem daunting, but breaking it down into manageable intervals helps. When graphing \(f(x) = |x| - |x-6|\), it is important to consider each interval separately, as the function's behavior changes across them. Ensuring accuracy in graphing such intervals is key:

• For \(x < 0\), the graph is a horizontal line at \(f(x) = 6\). Visualize it as a constant value, always hitting the same point on the y-axis regardless of x.
• In the next interval \(0 \leq x < 6\), you plot a linear increase starting at \(f(0) = -6\) and moving up to \(f(6) = 6\). This section appears as a slant line moving upward diagonally, showing a proportional rise in outputs as x increases.
• For \(x \geq 6\), the graph returns to another horizontal line at \(f(x) = 6\), reinforcing the steadiness after a change, once the absolute value functions align positively.

To sketch such a graph accurately, mark the key transition points \(x = 0\) and \(x = 6\), clearly defining the boundaries of each interval. This structured approach not only aids in understanding the function's varying behavior but also the impact of piecewise divisions on the overall graph.