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Chapter 2: Problem 116

What does it mean if a function \(f\) is increasing on an interval?

Short Answer

Expert verified
A function \(f\) is said to be increasing on an interval if, for all pairs of numbers \(x\) and \(y\) in the interval, if \(x < y\), then \(f(x) 鈮� f(y)\). This means that as you progress from left to right in the interval, function values do not decrease.

Step by step solution

01

Defining Interval

An interval, in mathematics, is defined as the set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers \(x\) satisfying 0 鈮� \(x\) 鈮�1 is an interval which we call [0,1].
02

Defining Increasing Function

A function \(f\) is increasing on an interval if, and only if, for all pairs of numbers \(x\) and \(y\) in the interval with \(x < y\), then \(f(x) 鈮� f(y)\). That is, the value of the function at a larger input is greater than or equal to the value of the function at a smaller input.
03

Visualizing Increasing Function

If we're to draw the graph of the function, an increasing function would rise as we move from the left to the right. Hence, for every pair of points on the function curve, if the point to the right is at a higher position than the point on the left, then the function is increasing on the interval between those points.

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