Question 1178867: Determine which two functions are inverses of each other.
F(x)= 9x G(x)=(x)/(9) H(x)=(9)/(x)
Found 3 solutions by ewatrrr, Edwin McCravy, greenestamps:
Answer by ewatrrr(24785)   (Show Source):
You can put this solution on YOUR website!
Hi
Determine which two functions are inverses of each other:
F(x)= 9x & G(x)=(x)/(9)
F^-1(x):F(x)= 9x, exchange(x&y) x= 9y 0r y = x/9 checks.
G^-1(x): G(x)= x/9 exchange(x&y) x = y/9 0r y = 9x checks.
Wish You the Best in your Studies.
Answer by Edwin McCravy(20055)  (Show Source):
You can put this solution on YOUR website!
She's right except she didn't tell you that one of those, H(x) is ITS OWN
inverse! Yes, that's right! A function can be its own inverse.
She showed you how to find inverses by four steps, although she only
mentioned step 2, and assumed you knew the other 3 steps.
1. Replace F(x), G(x), or H(x), by y, respectively.
2. Interchange x and y.
3. Solve for y.
4. Replace y by F-1(x), G-1(x), or H-1(x), respectively.
Let's follow those 4 steps with H(x)=9/x
H(x)=9/x
1. y=9/x
2. x=9/y
3. xy=9
y=9/x
4. H-1=9/x
How about that! H(x) and H-1(x)=9/x both equal 9/x, so
that means H(x) is ITS OWN inverse!
She also didn't show you what the graph of a function and its inverse look
like graphically.
I will draw F(x)=9x in blue, G(x)=x/9 in green, on the same set of axes and
also a dotted graph in red of y=x, which is a line that goes 45° through the
origin. It's often called the "IDENTIty line" because its equation, y=x,
shows that it is the case where y and x are IDENTIcal.
Notice that F(x) and G(x), which are INVERSES are REFLECTIONS of EACH OTHER
in (or across) the dotted IDENTIty line.
Any function's graph and its inverse's graph are always REFLECTIONS of EACH
OTHER in (or across) the IDENTIty line.
Now let's draw the graph of H(x) in green, and also the identity line y=x:
See how the graph is ITS OWN REFLECTION in (or across) the dotted
IDENTITY LINE? H(x) is ITS OWN INVERSE!
Edwin
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
Interchanging x and y in a given function and solving for the new y is a good formal mathematical method for finding inverses; but it is overkill for a simple example like this.
An informal way to find the inverse of many relatively simple functions is to think in terms of an inverse "undoing" what the function does. That is, two functions are inverses of each other if, when you take an input and operate on it with one function and then the other, you get back to the original input.
That informal way of thinking makes this problem easy.
The function F(x) says "multiply the input by 9".
The function G(x) says "divide the input by 9".
The function H(x) says "divide 9 by the input".
Which two of those functions "undo" each other? Stated differently, for which two of those functions is it true that taking an input, applying one function and then the other, gets you back to the original input?
Clearly the answer is "multiply by 9 and then divide by 9" (or "divide by 9 and then multiply by 9").
ANSWER: F(x) and G(x) are inverses of each other.
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