Question Video: Recognizing the Inverse of the Sine Function from a Table of Function Values Mathematics

Video Transcript

The given tables show some values of 𝑓 of 𝑥, 𝑔 of 𝑥, and ℎ of 𝑥. Which function corresponds to the function sin inverse of 𝑥?

So as the question says, we have three tables and they represent values of 𝑓 of 𝑥, 𝑔 of 𝑥, and ℎ of 𝑥, respectively. One of these functions 𝑓 of 𝑥, 𝑔 of 𝑥, or ℎ of 𝑥 is sin inverse of 𝑥, and we’re going to go through each option in turn to work out which one it is.

So first, we look at the table corresponding to 𝑓 of 𝑥 and we ask, “is 𝑓 of 𝑥 equal to sin inverse of 𝑥?” Well, if it is, then sin of 𝑓 of 𝑥 is equal to sin of sin inverse of 𝑥, and sin of sin inverse of 𝑥 is equal to 𝑥. So really, we’re asking if sin of 𝑓 of 𝑥 is equal to 𝑥. And we can test this using the values in the table. So we look at the first pair of values, where 𝑥 is negative root three by two and 𝑓 of 𝑥 is negative 𝜋 by three. And we ask for this pair of values if sin 𝑓 of 𝑥 is equal to 𝑥. So therefore, sin negative 𝜋 by three has to be equal to negative root three by two.

And either by knowing something about special angles or else using our calculator, we can see that this is in fact true. But of course, we need it to be true for the other values of 𝑥 and 𝑓 of 𝑥. And so we look at the second pair of values, 𝑥 is negative a half and 𝑓 of 𝑥 is negative 𝜋 by six. And so we have to ask, “is sin of negative 𝜋 by six equal to negative a half?” And of course, it is. So it’s looking good for 𝑓 of 𝑥, but we need to check all the values. So moving on, so looking at the next pair, we ask if a sin of zero is equal to zero, and of course it is. So moving on to the next pair of values, we ask if sin of 𝜋 by four is equal to a half, and of course it isn’t. Sin 𝜋 by four is equal to root two over two and not a half. And this means that for this pair of values, sin of 𝑓 of 𝑥 is not equal to 𝑥. And so 𝑓 of 𝑥 is not sin inverse of 𝑥. And so 𝑓 of 𝑥 is not our answer, and we move on to considering 𝑔 of 𝑥.

So it’s exactly the same process but for 𝑔 of 𝑥 instead of 𝑓 of 𝑥. To find out if 𝑔 of 𝑥 is equal to sin inverse of 𝑥, we ask if sin of 𝑔 of 𝑥 is equal to 𝑥, for all the values in our table. So considering the first pair of values, we ask if sin of negative 𝜋 by six is equal to negative root three over two. And the answer here is no; sin of negative 𝜋 by six is actually equal to negative a half. We’ve fallen at the first hurdle. Sin of 𝑔 of 𝑥 is not equal to 𝑥, in general. For this pair of values, it doesn’t hold, certainly. And so 𝑔 of 𝑥 is not sin inverse of 𝑥.

Now that we’ve eliminated both 𝑓 of 𝑥 and 𝑔 of 𝑥, we’ve only got one option left, ℎ of 𝑥. I’m not going to spend too much time on this but let’s just check that sin of ℎ of 𝑥 is equal to 𝑥, for all the values in the table. It turns out that sin of negative 𝜋 by two is negative one, sin of negative 𝜋 by six is equal to negative a half, sin of zero is zero, sin of 𝜋 by four is root two over two, and sin of 𝜋 by three is root three over two.

For all the values in our table, sin of ℎ of 𝑥 is equal to 𝑥. And so we conclude that ℎ of 𝑥 is sin inverse of 𝑥.