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CCSS.Math: ,

which function increases as x increases toward infinity and decreases as X decreases toward negative infinity so let's think about each of these constraints so first which function increases as x increases so this X increases toward infinity so X is going in that direction so first let's look at f of X over here so f of X as we get beyond this minimum point right over here as we increase our X f of X seems to be increasing so f of X seems to make the first constraint now let's think about G of X once we get past this a minimum point right over here as X gets larger and larger and larger as it approaches infinity G of X seems to be getting larger and larger and larger G of X is moving up so G of X also seems to make this first constraint now let's think about H of X as X moves towards infinity as X moves towards positive infinity H of X seems to be decreasing H of s seems to be decreasing so H of X does not even make the first constraint so our only two possibilities are now G of X and f of X so which of these decrease as X decreases toward negative infinity so let's think about that X decreasing toward negative infinity we're going to be going in that direction so first let's look at f of X so f of X it kind of goes up and down here but after we hit this little local minimum point this was a local maximum point this was a local minimum point over here not a global one as we move to the left of this lo of this local maximum point as we get smaller and smaller X's we see that the function is decreasing so it does seem to meet the second constraint it decreases as X decreases toward negative infinity so it meets that constraint now what about G of X after we have this minimum point and actually it looks like a global minimum point after we hit this minimum point right over here as X decreases toward negative infinity G of X seems to be increasing not decreasing so G of X does not meat does not meet the second constraint so the only function that met both constraints seems to be f of X