Constant Function — Definition, Formula & Examples

Constant Function

A function of the form y = constant or f(x) = constant, such as y = –2.

Key Formula

f(x) = c

Where:

$f(x)$ = the output of the function for any input $x$
$c$ = a fixed real number (the constant value)
$x$ = any real number input

Worked Example

Problem: Let f(x) = 5. Evaluate f(0), f(3), and f(−100), and describe the graph.

Step 1: Substitute each input into the function. Since

f(x) = 5

for all

x

, every output is 5.

f(0) = 5, \quad f(3) = 5, \quad f(-100) = 5

Step 2: The graph of

f(x) = 5

is a horizontal line passing through every point whose

y

-coordinate is 5.

Step 3: The slope of any horizontal line is zero, so the rate of change is 0.

m = 0

Answer: Every output equals 5, and the graph is the horizontal line

y = 5

with slope 0.

Why It Matters

Constant functions serve as the simplest building blocks in algebra and calculus. Their derivative is always zero, making them the baseline case when you study rates of change. They also appear frequently as one piece of a piecewise function, representing intervals where a quantity stays fixed.

Common Mistakes

Mistake: Thinking a constant function has no graph or is not a "real" function.

Correction: A constant function is a valid function — every input maps to exactly one output. Its graph is a horizontal line, not a single point.

Related Terms

Function — General concept that a constant function is a special case of
Constant — The fixed value that defines the function
Horizontal Line — The shape of a constant function's graph
Slope — Always equals zero for a constant function
Linear Function — Constant functions are linear functions with slope 0