Piecewise-Defined Functions Revision Notes for VCE SSCE Mathematical Methods

Piecewise-Defined Functions

What are piecewise-defined functions?

A piecewise-defined function applies different mathematical rules to different parts of its domain. These functions are also known as hybrid functions.

Instead of using a single rule across the entire domain, piecewise-defined functions break the domain into separate intervals, with each interval having its own specific rule or formula. This allows us to model situations where behaviour changes at certain points.

infoNote

Piecewise-defined functions are particularly useful for modelling real-world situations where behaviour changes at certain thresholds, such as:

Tax brackets that apply different rates to different income levels
Shipping costs that change based on package weight
Pricing structures with volume discounts

Graphing piecewise-defined functions

To sketch a piecewise-defined function, you need to graph each piece separately over its specified domain interval.

Key steps:

Identify each rule and its corresponding domain interval
Sketch each piece individually, paying attention to the domain restrictions
Check the endpoints carefully - determine whether they should be included (closed circle) or excluded (open circle)
Ensure the overall graph shows all pieces correctly connected

chatImportant

Watch for whether the inequalities include the equals sign ( $\leq$ or $\geq$ ) or not ( $<$ or $>$ ). This determines whether endpoints are filled in or left open.

Common mistake: Forgetting to check endpoint inclusion can lead to incorrectly drawn graphs and wrong range determinations.

Worked example: Linear piecewise function

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Worked Example: Graphing a Linear Piecewise Function

Consider the function:

f(x) = \begin{cases} -x - 1 & \text{for } x < 0 \\ 2x - 1 & \text{for } 0 \leq x \leq 1 \\ \frac{1}{2}x + \frac{1}{2} & \text{for } x > 1 \end{cases}

Task: Sketch the graph and state the range.

Solution:

To create this graph:

First piece: For $x < 0$ , we use the rule $y = -x - 1$

This is a straight line with slope $-1$ and $y$ -intercept at $-1$

We graph this line only for $x$ -values less than $0$

At $x = 0$ , the function value would be $-1$ , but this point is not included (shown by an open circle)

Second piece: For $0 \leq x \leq 1$ , we use the rule $y = 2x - 1$

This is a straight line with slope $2$ and $y$ -intercept at $-1$

When $x = 0$ , we get $y = -1$ (this endpoint is included, shown by a closed circle at $(0, -1)$ )

When $x = 1$ , we get $y = 1$ (this endpoint is also included, shown by a closed circle at $(1, 1)$ )

Third piece: For $x > 1$ , we use the rule $y = \frac{1}{2}x + \frac{1}{2}$

This is a straight line with slope $\frac{1}{2}$ and $y$ -intercept at $\frac{1}{2}$

We graph this line only for $x$ -values greater than $1$

At $x = 1$ , the function value would be $1$ , but this point belongs to the previous piece (shown by an open circle here)

Range: By examining the graph, we can see that the lowest $y$ -value is $-1$ (which is included), and the function continues upward indefinitely. Therefore, the range is $[-1, \infty)$ .

Worked example: Mixed piecewise function

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Worked Example: Graphing a Mixed Piecewise Function

Consider the function:

f(x) = \begin{cases} -4 & \text{for } x < -4 \\ -\frac{1}{2}x^2 + 2 & \text{for } -4 < x \leq 0 \\ 2x + 6 & \text{for } 0 < x \leq 2 \end{cases}

Task: Sketch the graph and state the range.

Solution:

To create this graph:

First piece: For $x < -4$ , we use the rule $y = -4$

This is a horizontal line at $y = -4$

We draw this line only for $x$ -values less than $-4$

At $x = -4$ , there is an open circle at $(-4, -4)$ since this point is not included

Second piece: For $-4 < x \leq 0$ , we use the rule $y = -\frac{1}{2}x^2 + 2$

This is a downward-opening parabola

At $x = -4$ , the value is $y = -\frac{1}{2}(16) + 2 = -8 + 2 = -6$ (open circle at $(-4, -6)$ )

At $x = 0$ , the value is $y = 2$ (closed circle at $(0, 2)$ )

Third piece: For $0 < x \leq 2$ , we use the rule $y = 2x + 6$

This is a straight line with slope $2$

At $x = 0$ , the value would be $y = 6$ (open circle at $(0, 6)$ - note the discontinuity)

At $x = 2$ , the value is $y = 10$ (closed circle at $(2, 10)$ )

Range: This function has gaps in its $y$ -values. Looking at the graph:

The second piece covers values from just above $-6$ up to and including $2$ : $(-6, 2]$

The third piece covers values from just above $6$ up to and including $10$ : $(6, 10]$

Therefore, the range is $(-6, 2] \cup (6, 10]$ (using union notation to combine the intervals).

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Notice how this function has a discontinuity at $x = 0$ . The function "jumps" from $y = 2$ to $y = 6$ , creating a gap in the range. This is why we need union notation to express the range - it consists of two separate intervals.

Remember!

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Key Points to Remember:

Piecewise-defined functions use different rules for different parts of their domain - think of them as "switching" between different formulas at specified points
Always check the inequality symbols carefully: $<$ or $>$ means open circle (point not included), while $\leq$ or $\geq$ means closed circle (point included)
To sketch the graph, draw each piece separately over its specified domain interval
The range is found by examining all $y$ -values that the complete graph covers - look for gaps or discontinuities
When stating the range, use interval notation and union symbols ( $\cup$ ) to join separate intervals if needed

Piecewise-Defined Functions (VCE SSCE Mathematical Methods): Revision Notes