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# Comparing functions: x-intercepts

CCSS Math: HSF.IF.C.9

## 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.