What is Monotonically Decreasing Function?

S7-SA1-0052

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

A monotonically decreasing function is a function where as the input value increases, the output value either stays the same or decreases. It never increases. Think of it like walking downhill or on a flat path, but never uphill.

Simple Example

Quick Example

Imagine your mobile data balance. If you start with 10 GB and use some data, your balance will either decrease (if you use it) or stay the same (if you don't use it). It will never increase on its own. This shows a monotonically decreasing pattern.

Worked Example

Step-by-Step

Let's check if the function f(x) = 10 - x is monotonically decreasing for positive x values.

---Step 1: Understand the function. f(x) takes an input x, subtracts it from 10, and gives the output.

---Step 2: Pick two input values, x1 and x2, such that x1 < x2. Let's choose x1 = 2 and x2 = 5.

---Step 3: Calculate f(x1). f(2) = 10 - 2 = 8.

---Step 4: Calculate f(x2). f(5) = 10 - 5 = 5.

---Step 5: Compare the outputs. We have f(x1) = 8 and f(x2) = 5. Since 8 > 5, this means f(x1) > f(x2).

---Step 6: Repeat with another pair. Let x1 = 10 and x2 = 12. f(10) = 10 - 10 = 0. f(12) = 10 - 12 = -2. Here, 0 > -2, so f(x1) > f(x2).

---Step 7: Since for any x1 < x2, we find that f(x1) >= f(x2) (in this case, always f(x1) > f(x2)), the function is monotonically decreasing.

Answer: Yes, f(x) = 10 - x is a monotonically decreasing function.

Why It Matters

Understanding these functions is crucial in many fields like AI/ML to optimize algorithms, in Physics to describe decaying processes, and in Economics to model falling prices or demand. Engineers and data scientists use this concept daily to build efficient systems and analyze trends.

Common Mistakes

MISTAKE: Confusing monotonically decreasing with strictly decreasing. | CORRECTION: Monotonically decreasing means the function can stay constant for some interval, then decrease. Strictly decreasing means it MUST always decrease, never staying constant.

MISTAKE: Only checking a few points and assuming the function is monotonic everywhere. | CORRECTION: To prove a function is monotonically decreasing, you need to show that for any x1 < x2 in its domain, f(x1) >= f(x2). Checking just two points isn't enough for a formal proof.

MISTAKE: Mixing up increasing and decreasing conditions (e.g., thinking f(x1) < f(x2) for decreasing). | CORRECTION: For a monotonically decreasing function, if x1 < x2, then f(x1) must be greater than or equal to f(x2). The output value goes down or stays the same.

Practice Questions

Try It Yourself

QUESTION: Is the function f(x) = 5 for all x (a constant function) monotonically decreasing? | ANSWER: Yes, because as x increases, f(x) stays the same (it doesn't increase), satisfying the condition f(x1) >= f(x2).

QUESTION: Consider the function f(x) = x^2 for x in the interval [0, 5]. Is it monotonically decreasing? | ANSWER: No. For example, if x1=1 and x2=2, then f(1)=1 and f(2)=4. Here, f(x1) < f(x2), which means it's increasing, not decreasing.

QUESTION: A shopkeeper sells samosas. The price of a samosa is Rs 10. For every 5 samosas bought, he gives a discount of Re 1 on each samosa (so for 5 samosas, each costs Rs 9; for 10 samosas, each costs Rs 9). For any number of samosas below 5, the price per samosa is Rs 10. Is the 'price per samosa' function monotonically decreasing as the 'number of samosas bought' increases? | ANSWER: Yes. From 1 to 4 samosas, the price per samosa is Rs 10. At 5 samosas, it drops to Rs 9. It stays Rs 9 until 9 samosas. It never increases, only stays the same or decreases. So, it is monotonically decreasing.

MCQ

Quick Quiz

Which of the following functions is monotonically decreasing?

f(x) = x + 5

f(x) = x^2 (for x > 0)

f(x) = 10 - 2x

f(x) = |x|

The Correct Answer Is:

Option C, f(x) = 10 - 2x, is monotonically decreasing because as x increases, 2x increases, so 10 - 2x decreases. Options A and B are increasing for x > 0, and D decreases then increases.

Real World Connection

In the Real World

Think about the battery level of your smartphone as you use it throughout the day. If you don't charge it, the battery percentage will either stay the same (if not in use) or decrease (when in use). It never increases on its own. This 'battery level over time' is a real-world example of a monotonically decreasing function. Similarly, the amount of fuel in a car as you drive is also monotonically decreasing.

Key Vocabulary

Key Terms

MONOTONIC: Always moving in one direction (either increasing or decreasing) or staying constant. | DECREASING: Getting smaller in value. | FUNCTION: A rule that assigns exactly one output for each input. | DOMAIN: The set of all possible input values for a function. | RANGE: The set of all possible output values for a function.

What's Next

What to Learn Next

Great job understanding monotonically decreasing functions! Next, you should explore 'Monotonically Increasing Functions' to see the opposite behavior. After that, you can dive into 'Strictly Monotonic Functions' which are important for understanding when a function has an inverse.