What is Increasing and Decreasing Functions?

S7-SA1-0049

Grade Level:

Class 12

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

Definition

What is it?

An increasing function is like your mobile data usage going up when you watch a lot of videos – as one value (time) increases, the other value (data used) also increases. A decreasing function is the opposite: as one value increases, the other value decreases, like the amount of battery left on your phone while you're playing games.

Simple Example

Quick Example

Imagine the number of runs scored by Virat Kohli in a match over different overs. If the runs keep going up as the overs increase, it's an increasing trend. If the number of unsold samosas at a shop goes down as the day progresses, that's a decreasing trend.

Worked Example

Step-by-Step

Let's check if the function f(x) = 2x + 3 is increasing or decreasing.

1. Pick two values for x, say x1 = 1 and x2 = 2, such that x1 < x2.
---2. Calculate f(x1): f(1) = 2(1) + 3 = 2 + 3 = 5.
---3. Calculate f(x2): f(2) = 2(2) + 3 = 4 + 3 = 7.
---4. Compare f(x1) and f(x2): We see that f(1) = 5 and f(2) = 7. Since 5 < 7, it means f(x1) < f(x2).
---5. Because x1 < x2 led to f(x1) < f(x2), the function is increasing.

Answer: The function f(x) = 2x + 3 is an increasing function.

Why It Matters

Understanding increasing and decreasing functions helps scientists predict climate change patterns and engineers design efficient electric vehicles. In finance, it helps analysts see if stock prices are rising or falling, guiding important investment decisions. This concept is key to many exciting careers!

Common Mistakes

MISTAKE: Assuming a function is increasing just because its values are positive. | CORRECTION: The sign of the function's value doesn't matter; what matters is whether the output value increases or decreases as the input value increases.

MISTAKE: Only checking two points to determine if a function is increasing or decreasing over its entire domain. | CORRECTION: For a formal proof, you need to show it holds for any x1 < x2 in the domain, or use calculus by checking the sign of the derivative.

MISTAKE: Confusing a function that is increasing with one that is always going up steeply. | CORRECTION: An increasing function can increase slowly or quickly, as long as its values never go down as x increases.

Practice Questions

Try It Yourself

QUESTION: Is the function f(x) = -x + 5 increasing or decreasing? | ANSWER: Decreasing

QUESTION: For the function f(x) = x^2, is it increasing or decreasing for x > 0? | ANSWER: Increasing

QUESTION: Consider the function g(x) = 1/x. Is it increasing or decreasing for x > 0? Explain. | ANSWER: Decreasing. As x increases (e.g., from 1 to 2), 1/x decreases (from 1 to 0.5).

MCQ

Quick Quiz

Which of the following describes an increasing function?

As x increases, f(x) decreases.

As x decreases, f(x) increases.

As x increases, f(x) also increases.

The value of f(x) is always positive.

The Correct Answer Is:

An increasing function means that as the input (x) gets larger, the output (f(x)) also gets larger. Option C correctly states this relationship.

Real World Connection

In the Real World

In India, tracking the number of daily UPI transactions is a real-world example. If the number of transactions keeps rising day by day, it shows an increasing trend in digital payment adoption. Similarly, predicting monsoon rainfall patterns involves analyzing increasing or decreasing trends in atmospheric pressure or temperature.

Key Vocabulary

Key Terms

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 of a function | MONOTONIC: A function that is either always increasing or always decreasing | DERIVATIVE: A tool from calculus used to find the rate of change of a function

What's Next

What to Learn Next

Now that you understand increasing and decreasing functions, you're ready to explore 'Local Maxima and Minima'. This next concept builds on identifying where a function changes from increasing to decreasing, which is crucial for finding peaks and valleys in graphs.