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# Functions

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1. I can do (i) but not (ii). The answer is h^-1(x)=h(x) but I just don't understand it! Thanks!

2. Multiply on the left by

So

since (identity)
3. (Original post by rpnom)

I can do (i) but not (ii). The answer is h^-1(x)=h(x) but I just don't understand it! Thanks!
Just some alternative wording to the above:

We have . Now you can apply a function to both sides of an equality (if it is injective, for the rigorous) so, for example when you have you can apply the square function to both sides of the equality to get .

Now you have . If we apply the function (that is, the inverse function ) then we get .

But when we apply the inverse function to the function itself, they just cancel out. So for example, when we do the square root of a square, it cancels out. . They just cancel each other out, you see? Because squaring and square rooting are inverses of one another.

So when we apply we get back out.

In this case, our anything is . So we have since the h^-1 cancels the first h.

So this means that is really just .

Let's recap:

We apply the inverse function to both sides.

We get .

We simplify the LHS by noting that the inverse function of a function cancels out.

We get .
4. (Original post by Zacken)
Just some alternative wording to the above:

We have . Now you can apply a function to both sides of an equality (if it is injective, for the rigorous) so, for example when you have you can apply the square function to both sides of the equality to get .

Now you have . If we apply the function (that is, the inverse function ) then we get .

But when we apply the inverse function to the function itself, they just cancel out. So for example, when we do the square root of a square, it cancels out. . They just cancel each other out, you see? Because squaring and square rooting are inverses of one another.

So when we apply we get back out.

In this case, our anything is . So we have since the h^-1 cancels the first h.

So this means that is really just .

Let's recap:

We apply the inverse function to both sides.

We get .

We simplify the LHS by noting that the inverse function of a function cancels out.

We get .
Thank you!!!!
5. (Original post by rpnom)
Thank you!!!!
No worries.

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