Absolute value & piecewise functions: FAQ

Absolute value is a function that outputs the distance of any input from $0$0 on the number line. We often write the absolute value of a number with vertical bars on either side, like $|x|$vertical bar, x, vertical bar.

For the most basic absolute value function, $y=|x|$y, equals, vertical bar, x, vertical bar, the graph looks like a "V" shape with the vertex at the origin. As we modify the function (for example, by shifting it vertically or horizontally), the graph will change accordingly, but it will maintain the general "V" shape.

Practice with our Shift absolute value graphs exercise.

Practice with our Scale & reflect absolute value graphs exercise.

Practice with our Graph absolute value functions exercise.

Piecewise functions are functions that are defined in separate "pieces" for different intervals of the input. For example, we could define a piecewise function $f(x)$f, left parenthesis, x, right parenthesis like this:

This function will multiply an input by $2$2 for inputs less than or equal to $0$0, and multiply it by $3$3 and add $1$1 for inputs greater than $0$0.

Practice with our Evaluate piecewise functions exercise.

The graph of a piecewise function will depend on the specific function itself. However, a common feature is that the graph may appear "broken" or disjointed at the points where the function changes definitions. For the function $f(x)$f, left parenthesis, x, right parenthesis above, the graph will be two different lines that meet at the origin.

Practice with our Piecewise functions graphs exercise.

Both types of functions are used in a variety of contexts. Absolute value functions can be helpful when we want to measure distances or find the magnitude of a certain quantity. Piecewise functions are often used to model situations where different rules or formulas apply in different situations.