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CCSS.Math: , , , ,

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.