Graphing Piecewise Functions

2 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

Graph each piece only where its condition applies. Endpoint markers are part of the mathematics: an open circle excludes a point, while a filled point assigns the value there.

Method

At each boundary, record the left-hand behavior, the right-hand behavior, and the function value separately. They can be three different things. A function may be defined at a boundary without being continuous there.

Worked example

Let $f(x)=x+1$ for $x<1$ and $f(x)=3-x$ for $x\ge1$ . Draw the first line only left of $1$ and the second only on and right of $1$ . Both approach $2$ , and the second assigns $f(1)=2$ , so the graph joins continuously. The left slope is $1$ and right slope is $-1$ , so there is a corner.

Common mistake

Continuity does not require the same formula or slope on both sides. Conversely, differentiability requires more than the pieces simply meeting.

Check your understanding

Replace the second branch by $4-x$ . Is the new function continuous at $1$ ?

Show answer

No. The left limit is $2$ , but the right limit and value are $3$ .

Explore

Value versus limit

Try this. Compare a hole and a jump. In the defined-value mode, move f(0) to 0: continuity requires the value to equal the common limit.

BehaviorDefined value f(0)0

Both sides approach 0, but f(0) is undefined. The hole is removable.

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

A piecewise graph has y=x for x<1 and y=3 for x≥1. Which point is filled at x=1?

(1,3)

(1,1)

Both points

Hint 1 · Find a starting point

A filled point belongs to the branch that includes equality.

Hint 2 · Take the next step

The second branch uses ≥.

Show the reasoning

Answer: (1,3)

The value at 1 is 3; (1,1) is an open endpoint.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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