Piecewise Functions: Videos & Practice Problems

Understanding piecewise functions is essential in mathematics, as they consist of multiple equations defined for specific intervals of x values. To graph a piecewise function, identify the boundaries where the equations change, ensuring no overlap occurs. For example, if x is less than -1, use one equation, while for x greater than or equal to -1, use another. Evaluating these functions involves substituting x values into the appropriate equations, which may reveal discontinuities, such as jump discontinuities, indicating that the function is not continuous across its domain.

Piecewise Functions

Understanding the concept of a function is fundamental in mathematics, particularly when exploring piecewise functions. A piecewise function is defined by multiple equations, each applicable to specific intervals of the independent variable, typically denoted as x. This means that for different ranges of x, different equations will govern the output of the function.

To illustrate, consider a piecewise function defined as follows:

f(x) =

In this example, the function is split at x = -1. To graph this function, one must recognize that the left side of the graph (where x is less than -1) will follow the equation -x, while the right side (where x is greater than or equal to -1) will follow the equation x² - 4.

When graphing, it is essential to denote the boundaries correctly. For x < -1, an open circle is placed at (-1, -3) to indicate that this point is not included in that part of the function. Conversely, for x ≥ -1, a solid circle is used at the same point to show that it is included in the second equation. This creates a visual representation of a jump discontinuity, where the two pieces of the function do not connect smoothly.

Evaluating a piecewise function involves substituting specific x values into the appropriate equations. For example:

• To find f(-3), since -3 < -1, we use the equation -x:
  f(-3) = -(-3) = 3.
• For f(-1), since -1 ≥ -1, we use x² - 4:
  f(-1) = (-1)² - 4 = 1 - 4 = -3.
• To find f(2), since 2 ≥ -1, we again use x² - 4:
  f(2) = 2² - 4 = 4 - 4 = 0.

These evaluations can be confirmed by checking the graph, where the calculated points align with the plotted function. Understanding how to graph and evaluate piecewise functions is crucial for mastering more complex mathematical concepts.

Using the piecewise function below, evaluate $$$f\left(-1\right)$

• A. $-4$
• B. $4$
• C. $\pm \sqrt{3}$
• D. $2$

Answer: (B)

Using the piecewise function below, evaluate $f\left(5\right)$

• A. $130$
• B. $125$
• C. $120$
• D. $30$

Answer: (A)

Piecewise Functions Example 1

To graph a piecewise function, it is essential to identify the boundaries defined by the conditions of the function. In this case, the function is divided into three segments based on the values of \( x \): less than \(-2\), between \(-2\) and \(1\), and greater than or equal to \(1\).

First, establish vertical lines at \( x = -2 \) and \( x = 1 \) to delineate the sections of the graph. The first piece of the function is a constant value of \(-4\) for \( x < -2\). This means that on the graph, a horizontal line will be drawn at \( y = -4 \) extending to the left of the line at \( x = -2\). Since the condition is \( x < -2\), an open circle is placed at \( (-2, -4) \) to indicate that this point is not included in the function.

Next, for the interval \(-2 < x < 1\), the function is defined as \( f(x) = x + 1 \). This linear function intersects the y-axis at \( (0, 1) \) and has a slope of \( 1 \). The line will be drawn between the points where \( x = -2 \) and \( x = 1\). At \( x = -2\), a solid dot is placed because the function includes this point, while at \( x = 1\), an open circle is used since the function does not include this endpoint.

Finally, for \( x \geq 1\), the function is defined as \( f(x) = x^2 \). This is a parabola that opens upwards, starting at the point \( (1, 1) \) since \( 1^2 = 1 \). The parabola will continue to rise as \( x \) increases, and a solid dot is placed at \( (1, 1) \) to indicate that this point is included in the function.

In summary, the piecewise function consists of three distinct segments: a horizontal line at \( y = -4 \) for \( x < -2\) with an open circle at \( (-2, -4) \); a linear segment from \( (-2, -1) \) to \( (1, 2) \) with a solid dot at \( (-2, -1) \) and an open circle at \( (1, 2) \); and a parabola starting at \( (1, 1) \) for \( x \geq 1\) with a solid dot at this point. Understanding how to graph these segments and where to place open and closed circles is crucial for accurately representing piecewise functions.

Here’s what students ask on this topic:

What is a piecewise function and how is it defined?

A piecewise function is a function composed of multiple sub-functions, each defined over a specific interval of the domain. It is represented as:

$f\left(x\right)=\{\begin{array}{cc}f\_1\left(x\right)& if x < a\\ f\_2\left(x\right)& if a \le x < b\\ f\_3\left(x\right)& if x \ge b\end{array}$

Each sub-function applies to a specific interval of x values, ensuring no overlap between intervals.

How do you graph a piecewise function?

To graph a piecewise function, follow these steps:

1. Identify the intervals for each sub-function.
2. Draw vertical lines (walls) at the boundaries of these intervals.
3. Graph each sub-function within its respective interval, ensuring to use open or closed circles at the boundaries based on the inequality signs.
4. Check for discontinuities, such as jump discontinuities, where the function values do not match at the boundaries.

This method ensures that each piece of the function is accurately represented within its defined interval.

What is a jump discontinuity in a piecewise function?

A jump discontinuity occurs in a piecewise function when there is a sudden change in the function's value at a boundary point. This means the left-hand limit and the right-hand limit at the boundary do not match. For example, if:

$f\left(x\right)=\{\begin{array}{cc}-x& if x < -1\\ x^2& if x \ge -1\end{array}$

There is a jump discontinuity at x = -1 because the function values on either side of -1 do not match.

How do you evaluate a piecewise function at a specific x value?

To evaluate a piecewise function at a specific x value, follow these steps:

1. Determine which interval the x value falls into.
2. Use the corresponding sub-function for that interval.
3. Substitute the x value into the chosen sub-function and solve.

For example, if:

$f\left(x\right)=\{\begin{array}{cc}-x& if x < -1\\ x^2& if x \ge -1\end{array}$

To find f(-3), use the first sub-function: f(-3) = -(-3) = 3.

Can a piecewise function be continuous?

Yes, a piecewise function can be continuous if the function values match at the boundaries of the intervals. This means the left-hand limit and the right-hand limit at each boundary point must be equal. For example, if:

$f\left(x\right)=\{\begin{array}{cc}-x& if x < -1\\ x^2& if x \ge -1\end{array}$

To ensure continuity at x = -1, the function values from both sub-functions must be equal at x = -1.