How to Determine if a Function is Strictly Increasing: A Step-by-Step Guide

strictly Increasing

In mathematics, a function is said to be strictly increasing if, for any two numbers in the domain, the value of the function is strictly greater for the second number compared to the first number

In mathematics, a function is said to be strictly increasing if, for any two numbers in the domain, the value of the function is strictly greater for the second number compared to the first number. In simpler terms, this means that as the input values increase, the corresponding output values also increase and don’t repeat.

To determine whether a function is strictly increasing or not, you need to analyze its behavior and assess whether it meets the conditions of strict increase. Here’s a step-by-step guide on how to determine if a function is strictly increasing:

1. Start by selecting any two distinct values in the domain of the function. Let’s call them x₁ and x₂, where x₁ < x₂. 2. Evaluate the function at these two points; compute f(x₁) and f(x₂) using the given function. 3. Compare the values of f(x₁) and f(x₂). If f(x₁) < f(x₂), then the function is strictly increasing on the interval [x₁,x₂]. If f(x₁) > f(x₂), then the function is strictly decreasing on the interval [x₁,x₂]. If f(x₁) = f(x₂), then the function is not strictly increasing or decreasing.

4. Repeat steps 1-3 for different pairs of values in the domain.

If, after analyzing several pairs of values, it is consistently found that f(x₁) < f(x₂) for all valid pairs, then the function is strictly increasing across its entire domain. However, if there is even a single pair of values for which f(x₁) > f(x₂), then the function is not strictly increasing.

It’s important to note that for a function to be strictly increasing, it must be defined on an interval where x₁ < x₂. If the function has specific restrictions or a limited domain, make sure to consider those constraints when determining strict increase. Overall, determining whether a function is strictly increasing involves comparing the function's values at two different points in its domain to verify that the output values progressively increase as the input values do.

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