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ACT Prep: Tutoring Solution43 chapters | 384 lessons

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will learn what it means for a function to be increasing. We will also learn how to determine if a given function is increasing by observing its graph and working with its derivative.

Let's imagine we are out for a walk. We are walking up a hill, and we notice that our height above sea level is increasing as the amount of time we are walking is increasing. If we were to consider our height above sea level, *y*, as a function of the amount of time we are walking, *x*, we would say that *y* is increasing as *x* is increasing while we are on that hill. In mathematics, we would say that when we are walking uphill, our function is an **increasing function**.

There are functions that are always increasing. For example, imagine you are at the store and you are buying some baseballs that costs $3 each. Your total cost, call it *c*, is a function of how many baseballs you buy, call it *x*, and can be represented as *c*(*x*) = 3*x*. Notice that your total cost will always go up as the number of baseballs you buy goes up. This function is always increasing. When a function is always increasing, we call it a **strictly increasing function**.

We see why our walking example is not a strictly increasing function. Sometimes the road or trail we are walking on goes uphill, sometimes it goes downhill, and sometimes, it is flat, so this is not strictly increasing if we consider the whole time we are walking. Therefore, we say the function is increasing during the time intervals that we are walking uphill.

It is easy to understand that when there are downhill sections on our walk, our function is only increasing during certain time intervals of our walk. Namely, when we are going uphill. You may wonder what this means for a function when it is constant or in terms of our walking function, when the road is flat. In other words, what if we were to go for a walk and our route consisted only of flat sections and uphill sections with no downhill sections? When this is the case, we would still say that our function representing this walk is increasing, but we wouldn't say that it is strictly increasing. However, if we were to go on a walk that was strictly uphill with no downhill or flat sections, then we would say the function representing this walk is strictly increasing.

Let's explain this further by observing the graphs of functions.

It is easy to determine if a function is increasing by observing the graph of a function. When a function is increasing, the graph of the function is rising from left to right. Consider the function *f*(*x*) = 2^*x*. The graph of this function is shown below.

Notice that this graph is always rising from left to right. It is easy to see that the function is strictly increasing. Now consider the function *g*(*x*) = *x*^2 shown in the graph below.

We see that the graph is falling from left to right when *x*<0 and it is rising from left to right when *x*>0. Thus, we wouldn't call this function strictly increasing since it's not increasing everywhere. Instead, we would say that *g*(*x*) = *x*^2 is an increasing function when *x*>0.

When the graph of a function is always rising from left to right, it is a strictly increasing function. When it is always rising from left to right or flat, then it is an increasing function, but not a strictly increasing function. Lastly, when the graph is rising and falling from left to right on different intervals, then the function is only increasing on certain intervals.

When we are unable to observe the graph of a function, we can use the derivative of the function to determine if it is increasing. As a reminder, the derivative of a function is the rate of change of *y* with respect to *x* at a given point of a function. When we think about this, we notice that if the rate of change is positive, then *y* is increasing as *x* is increasing, and this is the definition of an increasing function. Therefore, if the derivative of a function is positive, then the function is increasing.

Let's consider our example of *g*(*x*) = *x*^2. The derivative of *g*(*x*) = *x*^2 is *g* ' (*x*) = 2*x*. Since we are multiplying *x* by the positive number 2, it is easy to see that *g* ' is positive when *x* is positive and negative when *x* is negative. This tells us that *g* is increasing when *x* is positive, just like we observed in its graph.

Now consider our baseball example represented by the function *c*(*x*) = 3*x*. The derivative of this function is *c* ' (*x*) = 3. We know that 3 is always positive, so what does this tell us? You guessed it, the function *c*(*x*) is always increasing, so it is a strictly increasing function. We can observe this by looking at the graph of *c*(*x*) = 3*x*.

A function is said to be increasing if *y* is increasing when *x* is increasing. When a function is always increasing, we say the function is strictly increasing. We've seen what the graph of a function looks like when it is increasing. When a function is increasing, its graph rises from left to right. When we are unable to observe the graph of a function, we can check the derivative of the function to determine if it is increasing. When a function's derivative is positive, the function is increasing. When we use all of this information together, we get a thorough understanding of what it means for a function to be increasing. You may never look at going for a walk the same again!

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ACT Prep: Tutoring Solution43 chapters | 384 lessons

- What is a Function: Basics and Key Terms 7:57
- Inverse Functions 6:05
- Applying Function Operations Practice Problems 5:17
- How to Compose Functions 6:52
- How to Add, Subtract, Multiply and Divide Functions 6:43
- What Is Domain and Range in a Function? 8:32
- Functions: Identification, Notation & Practice Problems 9:24
- Compounding Functions and Graphing Functions of Functions 7:47
- Understanding and Graphing the Inverse Function 7:31
- Polynomial Functions: Properties and Factoring 7:45
- Polynomial Functions: Exponentials and Simplifying 7:45
- Explicit Functions: Definition & Examples 7:36
- Function Operation: Definition & Overview 6:17
- Function Table in Math: Definition, Rules & Examples 5:53
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