Graphing Piecewise Functions

Piecewise Functions

How to graph the given piecewise functions: examples and their solutions.

Example 1

Graph the given function. y = 2x + 3 (x < 0), x - 1 (x >= 0)

Draw y = 2x + 3 (x < 0)
on the coordinate plane.

[x < 0] doesn't include [x = 0].

So draw an empty point
on the y-intercept: 3.

Slope-intercept form

Draw y = x - 1 (x ≥ 0)
on the coordinate plane.

[x ≥ 0] does include [x = 0].

So draw a full point
on the y-intercept: -1.

So this is the graph of the given piecewise function.

Example 2

Graph the given function. y = x^2 (x < 1), x (x >= 1)

Draw y = x2 (x < 1)
on the coordinate plane.

[x < 1] doesn't include [x = 1].

So draw an empty point on the graph
at [x = 1].

Quadratic function - Vertex form

Draw y = x (x ≥ 1)
on the coordinate plane.

[x ≥ 1] does include [x = 1].

So draw a full point on the graph
at [x = 1].

But the empty point is already on (1, 1).
So fill in the empty point.

The empty point and the full point
are at the same point: (1, 1).

So erase the filled point at (1, 1).

So this is the graph of the given piecewise function.

Example 3

Graph the given function. y = (x + 1)(x - 3)/(x - 3) (x not= 3), -2 (x = 3)

See the first case of the given piecewise function.

[x ≠ 3] means [x - 3 ≠ 0].

So cancel (x - 3)
on both of the numerator and the denominator.

Then y = x + 1.

Simplifying rational expressions

So the given piecewise function
can be written like this.

Draw y = x + 1 (x ≠ 3)
on the coordinate plane.

The line doesn't include the point at [x = 3].

So draw an empty point on the line
at x = 3.

y = -2 (x = 3)
means (3, -2).

So draw a full point at (3, -2).

So this is the graph of the given piecewise function.