Solving Inverse Functions: Help Understanding h(x) & g(x)

[lilbite]

Homework Statement

If: h(x) and g(x) are inverse functions, then g[h(3)] = h[g(3)] =

Homework Equations

my teacher has neglected, yet again, to teach us how to do this.. could someone please help me.. this is all he gave us.. no functions or anything else to plug the #'s into..

The Attempt at a Solution

 

[CompuChip]

Welcome to PF lilbite.

By definition, an inverse function of f is a function which, when composed with f, forms the identity.

More informally: it takes you back to where you were.

More formally:
Let $f: A \to B$ be a function. A function $g: B \to A$ is called an inverse when $f(g(x)) = g(f(x)) = x$ for all $x \in A$.

For example, the square root is an inverse of the function
$$f: [0, \infty[ \to [0, \infty[, f(x) = x^2$$
because $\sqrt{x^2} = (\sqrt{x})^2 = x$
for all x in the domain.

 

[lilbite]

ok.. math is obviously not my strong point.. so I am guessing g=h is not the answer.. honestly everything you just told me just flew over my head.. which is sinking fast.. is there any way you could put this in easier terms?

 

[CompuChip]

Let me give you an easier example first then.
Remember when you learned about multiplication and division?

Division is like the inverse operation of multiplication. If you take any number and multiply it by x (not equal to zero) and then divide by x, you will get the same number back. For example, (3 x 6) / 6 = 3 and (12 x 786123) / 786123 = 12.
In fact, it also works the other way around: you can first divide by a number and then multiply by it: (3 / 6) x 6 = 3 and (12 / 786123) x 786123 = 12 again.

Now suppose that f is a function which multiplies by 6. So f(x) = x * 6 (I am now switching to * for multiplication, to prevent confusion with the variable x). Let g be the function which divides by 6: g(x) = x / 6. Now in terms of f and g, what I just told you is simply that f(g(3)) = g(f(3)) = 3, and in fact for any x, f(g(x)) = g(f(x)) = x.
So if I first apply one of the functions to some number x, and then apply the other of the two to the result, I get my number x back. This is precisely what is meant by f and g being inverses of each other.

You might also want to read at least the first part of the Wikipedia page on inverse functions and maybe go through my earlier example again. Sooner or later you will have to get used to this math notation

 

[lilbite]

ok.. now this i get.. thank you.. :)

 

[Mentallic]

Thanks for this simplified explanation compuchip It really helped me out too.

Also, it's much more tolerable than:

More formally:
Let $f: A \to B$ be a function. A function $g: B \to A$ is called an inverse when $f(g(x)) = g(f(x)) = x$ for all $x \in A$.

eww..

 

[Mark44]

Homework Statement

If: h(x) and g(x) are inverse functions, then g[h(3)] = h[g(3)] =

Homework Equations

my teacher has neglected, yet again, to teach us how to do this.. could someone please help me.. this is all he gave us.. no functions or anything else to plug the #'s into..

The Attempt at a Solution

Given that h and g are inverse functions, g(h(x)) is not necessarily equal to h(g(x)). The two will be equal if and only if the number x is in the domain for both h and g.

For example, let g(x) = ln(x) and h(x) = e^x, two functions that are inverses of each other.

Although g(h(x)) = x for all real numbers x, h(g(x)) isn't always defined, such as for h(g(-1)) = e^ln(-1).

 

[lilbite]

my teacher doesn't think through stuff... and if i knew how to do math i would luv to prove your equation right.. but i don't know... oh and i found out the answer is 3...

 

[[Quadratic]]

edited until I think something through :)

 

[Mark44]

my teacher doesn't think through stuff... and if i knew how to do math i would luv to prove your equation right.. but i don't know... oh and i found out the answer is 3...

You don't have to prove my equation right--just understand it. Regarding your problem, there might have been more to it than you provided. In general, with f and g being inverse functions, it is not necessarily true that f(g(x)) = g(f(x)). Here is a counterexample to the statement in your problem that f(g(3)) = g(f(3)) for inverse functions f and g.

Let f(x) = ln(x - 3), and let g(x) = e^x + 3

f(g(3)) = 3, but g(f(3)) is not defined, since f isn't defined at 3.

In other words, with the problem that you presented, the answer is not 3 in all circumstances, and my counterexample shows why it isn't.

If f and g are inverse functions, and the number x is the domain of both functions, then it will always be true that f(g(x)) = g(f(x)) = x.