0. Functions

Piecewise Functions

0. Functions

Piecewise Functions: Videos & Practice Problems

Topic summary

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.

concept

Piecewise Functions

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Piecewise Functions Video Summary

Understanding functions is fundamental in mathematics, and one important type of function is the piecewise function. 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) = { -x, &

Problem

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

$-4$

$4$

$\pm \sqrt{3}$

$2$

Problem

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

$130$

$125$

$120$

$30$

example

Piecewise Functions Example 1

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Piecewise Functions Example 1 Video Summary

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 segment of the function is defined for $ x < -2 $ and is a constant function where $ f(x) = -4 $. This means that for all values of $ x $ less than $-2$, the graph will have a horizontal line at $ y = -4 $. Since the condition is strict ($ x < -2 $), an open circle is placed at $ (-2, -4) $ to indicate that this point is not included in the graph.

The second segment applies to $ -2 \leq x < 1 $ and is represented by the linear function $ f(x) = x + 1 $. This line intersects the y-axis at $ (0, 1) $ and will extend from the point where $ x = -2 $ (where it equals $-1$) to just before $ x = 1 $ (where it equals $2$). At $ x = -2 $, a solid dot is placed to indicate that this point is included, while an open circle is placed at $ x = 1 $ since the function does not include this endpoint.

Finally, for the segment where $ x \geq 1 $, the function is defined as $ f(x) = x^2 $. This is a parabolic function that opens upwards. The graph starts at the point $ (1, 1) $ since $ 1^2 = 1 $ and continues to rise as $ x $ increases. A solid dot is placed at $ (1, 1) $ to indicate that this point is included in the graph.

In summary, the piecewise function consists of a horizontal line at $ y = -4 $ for $ x < -2 $, a linear segment from $ (-2, -1) $ to just below $ (1, 2) $, and a parabolic curve starting at $ (1, 1) $ and extending to the right. The placement of open and solid dots is crucial for accurately representing the function's domain and range.

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0. Functions - Part 1 of 2

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0. Functions - Part 2 of 2

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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. These sub-functions are combined to form a single function, but they do not overlap. For example, a piecewise function can be defined as:

$f (x) = {\begin{matrix} - x & if x < - 1 \\ x^{2} - 4 & if x \geq - 1 \end{matrix}$

Each sub-function is applied to a specific interval of x values, ensuring no overlap occurs.

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How do you graph a piecewise function?

To graph a piecewise function, follow these steps:

Identify the intervals for each sub-function.
Draw vertical lines (walls) at the boundaries where the sub-functions change.
Graph each sub-function within its specified interval.
Use open or closed circles at the boundaries to indicate whether the endpoint is included or not.

For example, for the piecewise function:

$f (x) = {\begin{matrix} - x & if x < - 1 \\ x^{2} - 4 & if x \geq - 1 \end{matrix}$

Graph $- x$ for $x < - 1$ and $x^{2} - 4$ for $x \geq - 1$ .

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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 between two sub-functions. This means the y-values of the sub-functions do not match at the boundary, causing a 'jump' in the graph. For example, if a piecewise function is defined as:

$f (x) = {\begin{matrix} - x & if x < - 1 \\ x^{2} - 4 & if x \geq - 1 \end{matrix}$

There is a jump discontinuity at $x = - 1$ because the y-values of the sub-functions do not match at this point.

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

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

Determine which sub-function applies to the given x value based on the defined intervals.
Substitute the x value into the appropriate sub-function.

For example, consider the piecewise function:

$f (x) = {\begin{matrix} - x & if x < - 1 \\ x^{2} - 4 & if x \geq - 1 \end{matrix}$

To evaluate $f (- 3)$ , use the sub-function $- x$ because $- 3 < - 1$ . Thus, $f (- 3) = - (- 3) = 3$ .

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Can a piecewise function be continuous? If so, how?

Yes, a piecewise function can be continuous if the sub-functions are defined in such a way that there are no jumps or breaks at the boundaries where the sub-functions change. This means the y-values of the sub-functions must match at the boundary points. For example, consider the piecewise function:

$f (x) = {\begin{matrix} - x + 1 & if x < 2 \\ 3 x - 5 & if x \geq 2 \end{matrix}$