Problem 100 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

A function \(f\) is said to be increasing on an interval if for all pairs of numbers \(x\) and \(y\) in that interval, if \(x < y\) then \(f(x) < f(y)\). This means that as we move from left to right, the function values increase.

Step 1: Definition of an Increasing Function

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 \(f(x) < f(y)\). This means that as we move from left to right (i.e., as the \(x\) values increase), the function values, or \(f(x)\) values, also increase.

Step 2: Demonstration with an Example

Consider a simple function like \( f(x) = x^2 \) on the interval \([0, +\infty)\). It can be seen that for every two numbers \(x\) and \(y\) in \([0, +\infty)\) such that \(x < y\), the values of \(f(x)\) will be less than the values of \(f(y)\). Hence, the function \(f\) is increasing on the interval \([0, +\infty)\).

Step 3: Visual Representation

When you plot the function on a graph, an increasing function should always appear to rise from left to right. In the graph of \(f(x) = x^2\) for \([0,+\infty)\), the graph moves upward as we move from left to right.