Piecewise, Step & Absolute Value Functions

Q: How do I pick which piece to use?

Check which condition your input satisfies. The conditions partition the domain, so exactly one piece applies to any given x.

Q: Why is absolute value considered piecewise?

Because it is defined by two rules: $x$ for non-negative inputs and $-x$ for negative ones. Those two lines form the V.

Q: What makes a step function 'step'?

It stays constant over each interval and jumps to a new constant at the boundaries, tracing a staircase rather than a smooth curve.

A piecewise function uses different rules on different parts of its domain. To evaluate it you first decide which piece your input falls into, then apply that piece's rule.

Step functions and absolute value functions are famous piecewise examples — absolute value is just two lines meeting at a V. The whole skill is choosing the right piece.

Evaluating a piecewise function

Each piece comes with a condition, like $x < 0$ or $x \geq 2$ . Match your input to the condition it satisfies, then plug into that piece only.

The conditions never overlap, so exactly one piece applies to any input. Read the boundaries carefully — $\leq$ versus $<$ decides which piece owns the boundary value.

Absolute value and step functions

Absolute value $|x|$ is piecewise: it equals $x$ when $x \geq 0$ and $-x$ when $x < 0$ , producing the V-shaped graph shown below — the right arm is the line $y = x$ , the left arm the line $y = -x$ .

A step function like the greatest-integer function jumps from one flat level to the next, staying constant on each piece — the source of its staircase look.

Worked examples

Example 1: choosing the right piece

For $f(x) = x^2$ if $x < 0$ and $f(x) = 3x$ if $x \geq 0$ , find $f(-2)$ .

Is -2 less than 0? Yes

\text{use } x^2

Substitute

(-2)^2

Simplify

4

Answer: $4$

Example 2: absolute value as pieces

Evaluate $|{-5}|$ using the piecewise definition.

-5 is negative, so use -x

-(-5)

Simplify

5

Answer: $5$

Example 3: an input right on the boundary

For $f(x) = x^2$ if $x < 0$ and $f(x) = 3x$ if $x \geq 0$ , find $f(0)$ .

Is 0 less than 0? No

\text{first piece does not apply}

0 satisfies the second condition, so the boundary belongs to that piece

\text{use } 3x

Substitute

3(0) = 0

Answer: $0$

Try one yourself

Common questions

How do I pick which piece to use?

Check which condition your input satisfies. The conditions partition the domain, so exactly one piece applies to any given x.

Why is absolute value considered piecewise?

Because it is defined by two rules: $x$ for non-negative inputs and $-x$ for negative ones. Those two lines form the V.

What makes a step function 'step'?

It stays constant over each interval and jumps to a new constant at the boundaries, tracing a staircase rather than a smooth curve.

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Piecewise, Step & Absolute Value Functions

Evaluating a piecewise function

Absolute value and step functions

Worked examples

Example 1: choosing the right piece

Example 2: absolute value as pieces

Example 3: an input right on the boundary

Try one yourself

Common questions

Keep learning

Absolute Value Functions

Functions & Continuity

Key Features of Functions