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## Algebra (all content)

### Course: Algebra (all content)>Unit 7

Lesson 29: Comparing features of functions (Algebra 2 level)

# Comparing functions: x-intercepts

Given several functions represented in different forms (as a formula, as a graph, and as a table of values), Sal finds the one with that has no x-intercepts. Created by Sal Khan.

## Want to join the conversation?

• Can anyone easily explain the difference between a function and an equation?
(15 votes)
• Just to clear up how you answered your own question:
A function has an X and Y axis, exactly like an equation.
However, all functions are equations, yet an equation is not necessarily a function.
The difference is that for every X value, a function can only have one Y value, whereas an equation can have multiple Y values for a single X.
(26 votes)
• ok I understand slightly more then I did before watching this video comparing features of functions what I want to know is why is this important? As a coach how can I explain why it is necessary to understand this in a real world scenario?
(3 votes)
• Lets all calm down guys. Ann, for your question, a very common real life application of functions is miles per gallon for cars. If you input 10 gallons of gas into a car that gets 20 mpg, then your output is that the car will travel around 200 miles. If you graph this function, then it will pass the vertical line test just like a function! Hope this helped you.
(5 votes)
• what does y and x intercept mean
(1 vote)
• The y intercept is whatever the y is equal to when x=0

The x-intercept is whatever the x value or values is/are whenever y=0. A function can have more than one x-intercept. Some functions have no x-intercept.
(3 votes)
• f is a function defined on non-negative integers. A verbal definition of f is given below
If the remainder of x divided by 3 is 0, than f(x) =11
If the remainder of x divided by 3 is 1, than f(x) = -7
If the remainder of x divided by 3 is 2, than f(x) =2
I don't understand these definitions for f given above . Can anyone easily explain this function definition ?
(1 vote)
• The domain ( x values ) is non-negative integers.
So once we have a value for x, we divide it by 3 and look at its remainder. That will determine which y value we use ( either 11, -7, or 2 ). Then we plot that point.
So, if x is 0 (the first non-negative integer), 0 divided by 3 has no remainder, so the point to plot on the graph is ( 0, 11 ).
If x is 1 (the next integer), 1 divided by 3 has a remainder of 1, so the point to plot on the graph is ( 1, -7 ).
And so on. This graph will be of distinct points all lying in either quadrant 1 or 4 (and, of course, one point will be on the Y-axis).
Hope this helps!
(2 votes)
• What's the difference between a principal root and a square root?
(1 vote)
• The principal square root is the positive real number square root of a positive number. For example, the square root of 9 could be 3 or -3, but the principal square root could only be 3.
Hope this helps!
(2 votes)
• wait what are functions??

can someone explain please and dont make it complicated please...
(1 vote)
• A function is something that takes in an input (like a number) and then returns an output. So, for example, the function "f(x) = 3x" takes in the input "x", and returns an output, which is just 3 times x. So "f(2)" takes in the input "2", and then spits out "3*2", which is just "6".

I won't go into too much detail here, because there are plenty of videos on Khan Academy that can explain it better than I can. So I'll just leave you this link to a video teaching about functions:

https://www.khanacademy.org/math/algebra2/functions_and_graphs/function-introduction/v/what-is-a-function

Good luck!
(2 votes)
• let´s say that f is a function defined on all integers.
plus the definition of this function is:
If x is even, then f(x) = -1
If x is odd, then f(x) = 3
How can you tell that f(0) = - 1 using only the above information?
(1 vote)
• 0 is an even number. So, it fits in your rule for even numbers.
(2 votes)
• Is there any way to view this on moble not through the app?
(1 vote)
• All Khan Academy videos are hosted through YouTube and are public listed, so yes. Just search for them on the YouTube mobile page.
(1 vote)
• Hi all. Had a similar question just turn up in masteries. It wa asking me whether an ungraphed function was periodic, how on earth do I check that. I dont recall it popping up in algebra 2 yet. Thanks.
(1 vote)
• Yes, periodic function it covered in Advanced Equations and Functions. You can check it out when get into Algebra 2.
(1 vote)
• At Sal say that we don't use sqrt(-5) as x as it is a complex number and hence, not real.
So are there function graphs for complex numbers?
Why do we not take in complex numbers in function graphs? Is it because we can't find their value?
Thanks.
(1 vote)

## Video transcript

Which function has no x-intercepts? So an x-intercept is a place where the function intersects the x-axis. And what do we know about what's going on when something is on the x-axis? Well, if something is on the x-axis, then you could say the y value is 0. Or if y is equal to the function, you would say that the value of the function is 0. You have an x-intercept whenever the function itself is equal to 0. So essentially this is equivalent to saying, which function never equals 0? So let's see if any of these functions never equal 0. So let's look at this first function right over here. And let me write it right over here. So I have f of x is equal to x squared plus 5. So this is interesting. x squared is always going to be a non-negative number. It'll be 0 or greater. Even if x is a negative value this is going to be 0 or greater. And 5 is obviously positive. So this whole value or this whole expression, x squared plus 5, is always going to be greater than or equal to 5. So we could say f of x is always going to be greater than or equal to 5. So f of x is never going to be equal to 0. If you don't believe me, let's try it out another way. Let's set f of x equals 0 and figure out at which x that might be true. So we could say 0 is equal to x squared plus 5. Subtract 5 from both sides. You would get negative 5 is equal to x squared. And if you take the principal root of both sides, you get the principal root of negative 5 is equal to x. You could even have the positive and negative principal root of negative 5. But needless to say, if you're dealing just with real numbers, there is no real number that is the square root of negative 5. So f of x has no x-intercepts. So this right over here, it meets the criteria. And this right over here has no x-intercept. So let's see if these other ones have x-intercepts. So remember, you have an x-intercept if the value of the function is 0 at some point. And we see right over here, this function g of x is really defined with this table. And we see it does indeed equal 0. It happens to equal 0 when x equals 0. So it intersects the x-axis right over there. That's its x-intercept. Now let's look at this green function, h of x. Where does that intersect the x-axis? Well, that's visually more obvious. It intersects the x-axis right over here. h of x is 0 when x is equal to negative 6. So these last two functions have x-intercepts. This first one does not.