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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.

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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.