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

That is the definition of inverse functions. But the situation in your problem is different. You're given that f and g are inverse functions, but you're not given that the number x is in the domain of both functions. Nevertheless, you can still find a value for f(g(3)) and another for g(f(3)), but those two values will not be the same in general.
  • #1
lilbite
4
0

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

 
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  • #2
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 [itex]f: A \to B[/itex] be a function. A function [itex]g: B \to A[/itex] is called an inverse when [itex]f(g(x)) = g(f(x)) = x[/itex] for all [itex]x \in A[/itex].

For example, the square root is an inverse of the function
[tex]f: [0, \infty[ \to [0, \infty[, f(x) = x^2[/tex]
because [itex]\sqrt{x^2} = (\sqrt{x})^2 = x[/itex]
for all x in the domain.
 
  • #3
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?
 
  • #4
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 :wink:
 
  • #5
ok.. now this i get.. thank you.. :)
 
  • #6
Thanks for this simplified explanation compuchip :smile: It really helped me out too.

Also, it's much more tolerable than:
CompuChip said:
More formally:
Let [itex]f: A \to B[/itex] be a function. A function [itex]g: B \to A[/itex] is called an inverse when [itex]f(g(x)) = g(f(x)) = x[/itex] for all [itex]x \in A[/itex].
eww.. :biggrin:
 
  • #7
lilbite said:

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) = ex, 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)) = eln(-1).
 
  • #8
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...
 
  • #9
edited until I think something through :)
 
  • #10
lilbite said:
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) = ex + 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.
 

1. What is an inverse function?

An inverse function is a function that undoes the action of another function. In other words, if a function f(x) takes an input x and produces an output y, the inverse function will take that output y and return the input x. This is represented mathematically as f^-1(y) = x.

2. How do you find the inverse of a function?

To find the inverse of a function, you must switch the input and output variables and solve for the new output. For example, if the original function is y = 2x + 3, the inverse function would be x = 2y + 3. Then, you would solve for y to get the inverse function y = (x-3)/2.

3. How do you check if two functions are inverses of each other?

To check if two functions are inverses, you can use the composition of functions method. This involves plugging one function into the other and seeing if you get the original input back. If f(x) and g(x) are inverses, then f(g(x)) = x and g(f(x)) = x for all values of x.

4. What is the relationship between the graphs of a function and its inverse?

The graphs of a function and its inverse are reflections of each other across the line y = x. This means that any point (x,y) on the graph of f(x) will be reflected across the line to become (y,x) on the graph of f^-1(x).

5. How can inverse functions be useful?

Inverse functions are useful in a variety of real-world applications, such as in cryptography, where they are used to encode and decode messages. They are also important in calculus, where they are used to find the derivative of a function. Inverse functions can also be used to solve equations and model real-world situations.

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