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### Course: Get ready for AP® Calculus > Unit 5

Lesson 2: Intervals where a function is positive, negative, increasing, or decreasing# Worked example: positive & negative intervals

Finding the positive or negative intervals of a function from its graph. Created by Sal Khan.

## Want to join the conversation?

- What I understand is that the result returned by
`f(x)`

completely ignores the`y`

value. Which is obviously wrong, otherwise the right answer to this question is if the plot is anywhere to the left of`X=0`

.

So can you please clarify that to me?(17 votes)- The function of x {f(x)} at any given point would be a coordinate plotted on the y-axis. You should consider them interchangeable. Any output derived from an input plotted along the horizontal x-axis can be, and usually is, plotted along the vertical y-axis. If you want to learn how to do these problem sets, just ask yourself if it's below the x-axis or not. If it's below, then it's less than zero. If it is above, then it is greater than zero.(7 votes)

- I don't understand what "F" is? Is it a variable or is something I haven't learn't?(6 votes)
- "f" is the name of a function. f(x) means the function f evaluated at x.

For example, if f(x) = 3x - 5, then f(3) = 3(3) -5 = 4. The graph of function f in that case would include, among many other things, the point (3,4). Hope that helps.(12 votes)

- I don't understand what the "graph is below the x-axis" means.(6 votes)
- It means that the graph (or part of it) is underneath the horizontal line that is the X-axis. It includes all the negative values of f(x).(8 votes)

- This doesn't cover how to do the equations, such as -1 > x > -5.(5 votes)
- In the following exercise “Positive and negative intervals” I have an issue understanding the terminology used. Sometimes they refer in the question to the function as f(x) and in the graph tells you that it’s equal to y, pretty traditional, but sometimes they refer in the question to the function as f.

Is there a difference?

Especially when it comes to which axis you need to look at to see the intervals over which the function is positive or negative.

For instance they’ll say “select the interval where h is positive” but other times they’ll say “select the interval where g(x)<0”(4 votes)- So they are strongly related, but have different purposes. If you are talking about the function, you can use just the letter to distinguish it from other functions that may be in the same problem or you could say f(x). However, if you write it in the equation, you have to use f(x) which is read as f of x and means the function f in terms of x. So one is talking about the function and the other is expressing the function in an equation. Thus f, f(x) and y are all the same and are used for different purposes. So both of your quoted requests are the same and asking where domain (x) meets some criteria.(3 votes)

- LoL The function literally achieved transcendence(5 votes)
- What are the real life examples of one to one function?(4 votes)
- You mean a slope of one? Like the y=x? Think of any relationship where one value is always equal to another. If I can buy one candy bar from the girl scouts for one dollar, and I then plotted the function of number of candy bars bought and the price of that amount of candy bars, it would be a y=x. Being such a simple function, it seems trivial to graph it when one value is equal to another, but it's good practice for slope, and can help prepare for more difficult functions. What if I could get five pieces of gum for a dollar? What would that function be?

y=x5 (y is the dollar amount spent. For each dollar, you have five times as many pieces of gum.)(1 vote)

- How to write x>2 in interval notation?(3 votes)
- This may sound stupid, but is there a horizontal line check?, is it when it has no slope. (It's really late, and I cant remember).(2 votes)
- The easiest to determent if it is negative or positive on a graph is when it is on the left top hand corner it is (-,+) below that it is (-,-). The top right hand corner is (+,+) while below it is (+,-).(2 votes)

- I dont understand what this means 2<x<3. How do you know where that is on the graph?(1 vote)
- 2<x<3 can be broken into 2 parts:

2<x: 2 is less than x

x<3: x is less than 3

When you put these together you get:

2 is less than x and x is less than 3

This means x is an fractional or decimal value located between 2 and 3.

When looking at the graph, look at the x-axis for the value between 2 and 3. Then look at what is happening on the graph for than set of numbers.

Hope this helps.(2 votes)

## Video transcript

A function, f of x,
is plotted below. Highlight an interval where
f of x is less than 0. So f of x-- which is
really being plotted on the vertical axis
right over here-- x is the horizontal axis. f of x being less
than 0 really means that the graph is
below the x-axis. So the function is negative in
this interval right over here and this interval over here. So I could put this
anywhere right over here, or I could stick it
anywhere right over here. Let me stick it right over here. There we go. Got it right. Let's do a couple more. So the function
is plotted below. Highlight an interval where
f of x is greater than 0. So I could do this
area right over here where the function
is above the x-axis, or I could do this
area right over here where the function goes
way above the x-axis. Well, it even goes off the page. So let's stick it
right over here. Let's do one more. So highlight an interval where
f of x is greater than 0. Once again, I can do this
region right over here where the function
is above the x-axis, or over here where
it's above the x-axis. I'll do it here just for fun. There we go.