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Problem 58 Hard Difficulty

( Let $F(x)=\sin x, G(x)=x^{2},$ and $H(x)=x^{3} .$ Which, if any, of the following four composite functions have graphs that do not go below the $x$ -axis? First, try to answer without using a graphing utility, then use the graphing utility to check yourself. (You will learn more this way than if you were to draw the graphs immediately.)
$$\begin{array}{ll}y=G(F(x)) & y=F(G(x)) \\y=F(H(x)) & y=H(F(x))\end{array}$$



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Video Transcript

so you're given these three functions. Therefore, fats is equal to sign off X, then you have g affects. Is he two squared and then h of X is a go to execute. And then we need to analyze these four functions. So lets for start with why is equal to G off ffx Well, that's equal to sign off X squared and that's equal to sign square off X. Now when you take the square off any number we know that sign is between negative one and one and you take the swear off that sign, then it belongs. Be positive. So this does not go the X axis, So that's the 1st 1 Now let's take a look at the next one, which is why is equal to F off G off X. That's equal Koo Sign off X squid now as X values increase, we know that sign is a period of two pi, and it takes values from negative want to positive one. So this graph off dysfunction way go below the X axis positive values in the Y direction as well as connected values. Next we have why is equal to F H off X and that would be equal. Do sign off ex huge now, same thing as X goes through different values that can quote values from 0 to 2 pi And in that double, somebody values will be necked. So this function does go B X axis and then the last one we have is why is it going to each off f of X? And that would be a sign off. Ex cute. So now, since sign can take values between negative wanted positive one when the sign values negative and sign off, X cubed will also be negative. So the graph dysfunction also goes below the X axis.

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