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CCSS.Math:

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 zero or if y is equal to the function you would say that the value of the function is zero you have an x-intercept whenever the function itself is equal to zero so essentially this is equivalent to saying which function never equals zero so let's see with if any of these functions never equal zero 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 value 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 what which X that might be true so though 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 unless you're dealing with if you're dealing just with real numbers there is no real number that is a square that is the square root of negative 5 so f of X has no x-intercepts so this right over here it meets the criterion this right over here has no x-intercepts 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 zero it happens to equal zero when x equals zero 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 zero when X is equal to negative six so these last two functions have x-intercepts this first one does not