Piecewise Function: Graph

y = 2x + 3
(x < 0)

So draw y = 2x + 3
on the left side of x = 0.

Slope-Intercept Form

x < 0 does not include x = 0.
So draw an empty circle at x = 0.

y = x - 1
(x ≥ 0)

So draw y = x - 1
on the right side of x = 0.

x ≥ 0 does include x = 0.
So draw a full circle at x = 0.

This is the graph of the given piecewise function.

y = x²
(x < 1)

So draw y = x²
on the left side of x = 1.

Quadratic Function: Vertex Form

x < 1 does not include x = 1.
So draw an empty circle at x = 1.

y = x
(x ≥ 1)

So draw y = x
on the right side of x = 1.

x ≥ 1 does include x = 1.
So draw a full circle at x = 1.

Both graphs meet at the same point
when x = 1.

So remove the circle
and connect the graphs.

This is the graph of the given piecewise function.

See (x + 1)(x - 3)/(x - 3).

When x ≠ 3,
x - 3 ≠ 0.

So you can cancel (x - 3) factors.

Then
(x + 1)(x - 3)/(x - 3) = x + 1.

Simplify a Rational Expression

Rewrite the given piecewise function.

Change
y = (x + 1)(x - 3)/(x - 3)
to
y = x + 1.

And write
y = -2
(x = 3).

Graph this piecewise function.

y = x + 1
(x ≠ 3)

So draw y = x + 1.
And draw an empty circle at x = 3.

The empty circle means
the y value is not on the line y = x + 1.

y = -2
(x = 3)

So draw a point on (3, -2).

This is the graph of the given piecewise function.