Piecewise Functions

Front

What is a piecewise function?

Back

A piecewise function is a function that is defined by different expressions or formulas for different intervals of the input variable.

Front

How do you evaluate a piecewise function?

Back

To evaluate a piecewise function, determine which interval the input value falls into and then use the corresponding expression to find the output.

Front

What is the general form of a piecewise function?

Back

The general form is: @@f(x) = \begin{cases} expression_1 & \text{if } condition_1 \\ expression_2 & \text{if } condition_2 \end{cases}$$

Front

What does the notation @@f(x) = \begin{cases} -x + 4 & \text{if } x < 2 \\ 2x - 6 & \text{if } x \geq 2 \end{cases}$$ represent?

Back

This notation represents a piecewise function where the output is determined by the value of x: it uses -x + 4 for x values less than 2 and 2x - 6 for x values greater than or equal to 2.

Front

How do you graph a piecewise function?

Back

To graph a piecewise function, plot each piece of the function according to its defined interval, ensuring to use open or closed circles to indicate whether endpoints are included.

Front

What is the output of the piecewise function @@f(x) = \begin{cases} x - 1 & \text{if } x \geq 0 \\ -2x + 1 & \text{if } x < 0 \end{cases}$$ when x = 3?

Back

The output is 2, since 3 is greater than or equal to 0, we use the expression x - 1.

Front

What is the output of the piecewise function @@f(x) = \begin{cases} x - 1 & \text{if } x \geq 0 \\ -2x + 1 & \text{if } x < 0 \end{cases}$$ when x = -2?

Back

The output is 5, since -2 is less than 0, we use the expression -2x + 1.

Front

What is the significance of open and closed intervals in piecewise functions?

Back

Open intervals indicate that the endpoint is not included in the function, while closed intervals indicate that the endpoint is included.

Front

How can you determine if a piecewise function is continuous?

Back

A piecewise function is continuous if the limits from both sides of a point match the function's value at that point.

Front

What is the domain of a piecewise function?

Back

The domain of a piecewise function is the set of all possible input values (x) for which the function is defined.