Increasing Function

Q: Can a function be both increasing and decreasing?

Not on the same interval. However, a function can be increasing on one interval and decreasing on another. For instance, f(x) = x² is decreasing on (−∞, 0) and increasing on (0, ∞). A constant function is technically neither increasing nor decreasing.

Q: How do you tell if a function is increasing from its graph?

Trace the graph from left to right. If the curve goes upward over some interval, the function is increasing there. Equivalently, if every point to the right is higher than every point to the left within that interval, the function is increasing on it.

Increasing Function

A function with a graph that goes up as it is followed from left to right. For example, any line with a positive slope is increasing.

Note: If a function is differentiable, then it is increasing at all points where its derivative is positive.

See also

Decreasing function

Key Formula

\text{A function } f \text{ is increasing on an interval if, for all } a \text{ and } b \text{ in that interval:}

a < b \implies f(a) < f(b)

\text{Derivative test: } f \text{ is increasing where } f'(x) > 0

Where:

$f$ = The function being analyzed
$a, b$ = Any two input values in the interval, with a less than b
$f'(x)$ = The derivative of f at x; when positive, the function is increasing at that point

Worked Example

Problem: Determine whether the function f(x) = 2x + 3 is increasing, and verify using two specific input values.

Step 1: Pick two values where a < b. Let a = 1 and b = 4.

a = 1, \quad b = 4

Step 2: Compute f(a) and f(b).

f(1) = 2(1) + 3 = 5, \quad f(4) = 2(4) + 3 = 11

Step 3: Check whether f(a) < f(b).

5 < 11 \quad \checkmark

Step 4: Confirm with the derivative. The derivative of f(x) = 2x + 3 is f'(x) = 2, which is positive for all x.

f'(x) = 2 > 0 \text{ for all } x

Answer: f(x) = 2x + 3 is an increasing function on its entire domain because its slope (derivative) is always positive.

Another Example

Unlike the first example (a linear function that is increasing everywhere), this cubic function is only increasing on certain intervals, showing that many functions increase on some parts of their domain and decrease on others.

Problem: Find the intervals where f(x) = x³ − 3x is increasing.

Step 1: Find the derivative of f.

f'(x) = 3x^2 - 3

Step 2: Set the derivative equal to zero to find critical points.

3x^2 - 3 = 0 \implies x^2 = 1 \implies x = -1 \text{ or } x = 1

Step 3: Test the sign of f'(x) in the three intervals created by the critical points.

f'(-2) = 3(4) - 3 = 9 > 0, \quad f'(0) = -3 < 0, \quad f'(2) = 9 > 0

Step 4: Identify where f'(x) > 0. The function is increasing on the intervals where the derivative is positive.

f \text{ is increasing on } (-\infty, -1) \text{ and } (1, \infty)

Answer: f(x) = x³ − 3x is increasing on (−∞, −1) and (1, ∞).

Frequently Asked Questions

What is the difference between increasing and strictly increasing?

A strictly increasing function requires f(a) < f(b) whenever a < b — the output must genuinely go up. A non-strictly increasing (sometimes called non-decreasing) function only requires f(a) ≤ f(b), allowing flat sections where the output stays the same. In most algebra and precalculus courses, 'increasing' means strictly increasing unless stated otherwise.

Can a function be both increasing and decreasing?

Not on the same interval. However, a function can be increasing on one interval and decreasing on another. For instance, f(x) = x² is decreasing on (−∞, 0) and increasing on (0, ∞). A constant function is technically neither increasing nor decreasing.

How do you tell if a function is increasing from its graph?

Trace the graph from left to right. If the curve goes upward over some interval, the function is increasing there. Equivalently, if every point to the right is higher than every point to the left within that interval, the function is increasing on it.

Increasing Function vs. Decreasing Function

	Increasing Function	Decreasing Function
Definition	Output rises as input increases: a < b implies f(a) < f(b)	Output falls as input increases: a < b implies f(a) > f(b)
Derivative sign	f'(x) > 0	f'(x) < 0
Graph behavior	Goes up from left to right	Goes down from left to right
Linear example	y = 3x + 1 (positive slope)	y = −2x + 5 (negative slope)

Why It Matters

Identifying where a function increases or decreases is central to finding maximums, minimums, and sketching accurate graphs — skills tested throughout algebra, precalculus, and calculus. In calculus, the first derivative test uses intervals of increase and decrease to classify critical points. Beyond math class, increasing functions model real-world growth such as population over time, distance traveled at constant speed, or account balances earning interest.

Common Mistakes

Mistake: Saying a function is increasing just because f(b) > f(a) for one pair of points.

Correction: A function is increasing on an interval only if f(a) < f(b) for every pair where a < b in that interval. One pair of points is not enough — you need to verify the condition holds throughout the entire interval, or use the derivative test.

Mistake: Concluding a function is increasing everywhere because its derivative is positive at a single point.

Correction: A positive derivative at one point means the function is increasing at that point, not on the whole domain. You must find all intervals where f'(x) > 0 to determine every region of increase.

Related Terms

Decreasing Function — Opposite behavior: output falls as input rises
Function — The general concept that increasing/decreasing describes
Derivative — Used to test whether a function is increasing
Slope of a Line — Positive slope means a linear function is increasing
Differentiable — Condition needed to apply the derivative test
Graph of an Equation or Inequality — Visual tool for identifying increasing intervals
Positive Number — Derivative must be positive for function to increase