math

Piecewise Function

A function defined by multiple sub-functions, each applying to a certain interval of the independent variable.

General Form of Piecewise Function

f(x) = { f1(x) if x < a, f2(x) if a ≤ x < b, f3(x) if x ≥ b }.

Example of a Piecewise Function

f(x) = { x^2 if x < 0, 2x + 1 if 0 ≤ x < 3, 5 if x ≥ 3 }.

Graphing Piecewise Functions

Graph each piece separately, using open or closed circles to indicate included endpoints.

Linear Function

A function expressed in the form f(x) = mx + b, where m is the slope and b is the y-intercept.

Slope (m)

Indicates the steepness and direction of a line, calculated as m = (y2 - y1) / (x2 - x1).

Y-Intercept (b)

The point where the line crosses the y-axis, occurring when x = 0.

Graphing Linear Functions

Start at the y-intercept (b) and use the slope (m) to find other points on the line.

Key Differences between Piecewise and Linear Functions

Piecewise functions can have multiple definitions and appear segmented, whereas linear functions have a constant slope and produce straight lines.

Continuity in Piecewise Functions

Check for continuity at boundaries between different sub-functions.

Domain and Range in Functions

Identify the domain and range for both piecewise and linear functions.