Comparing functions: shared features

Which of the features are shared by f of x and g of x? Select all that apply. So they give us f of x as being defined as x to the third minus x. And they define g of x, essentially, with this graph. So what are our options? So the first one is that they are both odd. So just by looking at g of x, you can tell that it is not odd. The biggest giveaway is that an odd function would go through the origin. G of zero would have to be equal to zero. If you want to go straight to the definition of an odd function, g of x would have to be equal to the negative of g of negative x. So for example, g of 3 looks like it is 4. g of 3 is equal to 4. In order for it to be odd, g of negative 3 would have to be equal to negative 4. But we see that g of negative 3 is not equal to negative 4. So this one is definitely not odd. So this statement can't be true. They both can't be odd. So that's not right. They share an x-intercept. So g of x only has one x-intercept. It intersects the x-axis right over here at x equals negative 3. Now let's think about the x-intercepts of f of x. And to do that, we just need to factor this expression, f of x is equal to x to the third minus x, which is the same thing if we factor an x out of x times x squared minus 1. X squared minus 1 is the difference of squares. So we could rewrite that as-- so we'll write our x first, this x. And then x squared minus 1 is x plus 1 times x minus 1. So when does f of x equal 0? Well, f of x is equal to 0 when x is equal to 0. When x is equal to 0, that would make this entire expression 0. When x is equal to negative 1, that would make this term, and thus the entire expression, 0. And when x is equal to positive 1, that would make this last part zero, which would make this entire product 0. So here are the zeroes of f of x, and none of these coincide with the zeroes of g of x. So they don't share an x-intercept. They have the same end behavior. Now this is interesting. This is saying what's happening as x gets really, really, really, really large, or as x gets really, really, really, really, really small. So we could just think about it right over here. As x gets really, really, really, really, really large, this x to the third is going to grow much faster than this x term right over here. So as x grows really, really, really, really large, f of x is going to grow really, really, really large. So the graph-- and I don't know exactly, to see if I could plot a couple of other points-- but the bottom line is f of x is going to approach infinity as x approaches infinity, or f of x approaches infinity as x approaches infinity, or as x gets larger and larger and larger. And then what happens as x gets smaller and smaller and smaller? If we have really small values of x-- so really negative values of x, I should say-- once again, this, right over here, is going to dominate. So f of x is going to become really negative. So f of x is going to approach negative infinity, as x approaches negative infinity. And that is the same behavior of g of x. As x approaches a really large value, g of x approaches a really large value, maybe not as fast as f of x, but it still approaches it. And likewise, as x decreases, so does g of x decrease. It doesn't decrease as fast as f of x might, but it's still going to decrease. So they do seem to have the same end behavior, at least based on the way that we thought about it just now. Now the last option is they have a relative maximum at the same x value. So we have to think about what the maximum points are. Well, actually, we already know that this is not true, because g of x has no relative maximum points. In order to have a maximum point, you would have to do something like this. This, right over here, would be a relative maximum, or, you could say, a local maximum point. It's larger than all of the points around it, but eventually the function does surpass it. But this, right over here, has no local maxima, or relative maxima, or little bumps in it. g of x doesn't have any. So they can't have relative maximum at the same value. So this, also, is not an option.