Question

How would I solve this type of equation?
Find the function f(x) - x + 9, find (f o f-1)(5)

A. 14
B. 5
C. 25
D. -5

Answer 1

Hello, Yana!
Here you have the composition of function f(x) with itself. This means that f(x) is the input to f(x): f(f(x)). Do that algebraically first. When done, let x=5.

Answer 2

Ok. So would it be:
f(x)-5+9?

Answer 3

Yana, you haven't yet taken the composition f(f)x)), so, no.

Try again. If you define f(x) as f(x)=-x+9 (check that, please), then

f( f(x) ) = f(-x+9) = -(-x+9)+9=??

Answer 4

just simplify the algebra here.

Answer 5

Ok. You want me to show you what I got?

Answer 6

You bet I do!

Answer 7

Ok. I'm a little confused. What do you mean by taking out the composition?

Answer 8

Still facing problem??

Answer 9

Yes. I do. Sorry couldn't reply earlier.

Answer 10

The problem is, I don't know how to set up the equation.

Answer 11

Where exactly??

Answer 12

Okay, it is okay, you should not be worried instead you must be happy that today you are going to add one more new thing to your knowledge.. :)

Answer 13

This part:
(f o f-1)(5) <-----Is that where I'm supposed to plug in, f(x) - x + 9?? That's where I'm confused.

Answer 14

Haha, I hope I understand this:D

Answer 15

Is that f^{-1} there??

Answer 16

Yes.

Answer 17

Firstly, I want to tell you about function somewhat..

Answer 18

One point that may help solve this problem quickly is:
If f(x) is a function and g(x) is its inverse then, f(g(x)) = x and g(f(x)) = x

Answer 19

Suppose, you have f(x) = x + 3, here f(x) means a function of x, as you can see here,

f(x) value depends on x (3 is constant there)..

Answer 20

I mean to say that: f(x) = x + 3 If I put x = 1,2,3 and so on, f(x) value will also change accordingly..

Here in f(x), x is basically called argument...
At the starting, you must have confusion, but as we proceed, you can easily get what we are doing..

Answer 21

Ok, yea. It's starting to make sense:D

Answer 22

One more thing in advance I want to tell you that :

f(x) = x+3

Now if I change my argument from x to (x+9) then where are you seeing x earlier, just replace it with (x+9)

So, I am replacing all x with (x+9)

Look it here, we get as:

f(x+9) = (x+9) + 3

Look it very carefully, are you getting it?? Look 100 times and say are you getting it because this is the point where you will be getting confused I suppose..

Answer 23

Yes!!!!!!!!!!!!!!!!!!!!!!!!! That's the part!!!!!!!!!!!!!!!!!!!!! Wow!!!! Youre such a great explainer!! By the way..are you a teacher? Or somethin like that?

Answer 24

Are you a student??

Answer 25

Yes.

Answer 26

I will turn 90 this 5th Of August.. :)

Answer 27

Wow!!! Ok. Anyways. Back on track:) And Happy Early Birthday:D

Answer 28

I think you will oppose that, but anyway, back on track now and don't distract me.. :P

Answer 29

Haha:) Ok. I won't hopefully:D

Answer 30

So getting that part??

Answer 31

Now, some more you have to learn before proceeding to our question..

f(x) can be written as y.. Okay??

So, f(x) = x+3 or y = x+3 both are one and the same thing..

Answer 32

Ok. Yes. I'm understanding it soo far.:D

Answer 33

Now, I want to know if you know how to know I mean how to find Inverse of f(x) ??

Answer 34

Yes. I think so.

Answer 35

Should I give an example or is it okay??

Answer 36

Give an example. I think that would be better:D

Answer 37

So, let us take a simple function here to understand how to find inverse..

Let us take f(x) = x+5

As I said, you can write f(x) as y too, so : y = x+ 5

Now First step would be to convert this expression in terms of x..

So, subtracting 5 from both sides, y - 5 = x, (this is in now terms of x)

Now second step is to just interchange x with y and y with x..

So, y = x - 5, and this y is here \(f^{-1}(x) = x-5\)

Check it till here if I am not wrong then..

Answer 38

Is it okay or somewhat hard to understand??

Answer 39

Umm..Will I be able to do it similarily with this one too?
f(x) - x + 9,

Answer 40

Yeah why not...

Answer 41

So, let us do first this only, okay??

Answer 42

Ok:) Can I show you what I have done by the way you have it and tell me if it's right or wrong?

Answer 43

Yeah, that will be better.. I am spectator now, Go Ahead.. :P

Answer 44

Hahha:D Ok. Umm. I'm not sure if I got this right, but I have:
\[f ^{-1}(x)=x-9\]

Answer 45

Wait, wait sorry to interrupt in between, In question: f(x) = x+9 no??

Answer 46

I mean in between that's equal to sign no??

Answer 47

What do you think, have you found correctly for inverse of f(x) ??? YES or NO??

Answer 48

There is no equal sign.

Answer 49

you mean f(x) - x + 9 ???

Answer 50

Yes. That's what I have above in my question.

Answer 51

Yana, this will create a problem, just look this problem once again, for sure f(x) = x+9 must be correct..

Answer 52

I have it both ways. The one that we're talking about and
f(x)=x+9. Lets just do \(\huge{\leftarrow}\)

Answer 53

I did not get you??

Answer 54

So, back to question, And yes the inverse you have found is correct (as I am the teacher here, just kidding), Good, you are learning something.. :)

Answer 55

Hey, I guess you are not writing a Novel??

Answer 56

I have to solve \(\huge{f(x) - x + 9}\)
and
\(\huge{f(x)=x+9}\)
They are totally two different questions.
For the second one I have:
\(\huge{f(x)=x-9}\)

Answer 57

Hhaa. No. I wasn't.

Answer 58

What does - sign mean there??

Answer 59

Negative.

Answer 60

Then I think equal to sign is missing there..

You have two question as : f(x) = -x + 9

and f(x) = x+9, these two will make sense..

Answer 61

Lets just stick with:
\(\huge{f(x)=x+9}\)

Answer 62

Right..

Answer 63

Yana: Apologies for not having stuck around to help you finish this problem.

I can't help but wonder if the f(x) - x + 9 you've typed in should be f(x) = -x + 9. Would you please check that?

Answer 64

As you have found its inverse, so now we should enter the territory of desired result..

Answer 65

@mathmale you can start it from here if you want, I have no problem... :)

Answer 66

To get anywhere with this problem, we MUST have f(x) definied somewhere. I don't see that. f(x) - x + 9 is not a definition of a function; f(x)=-x+9 is.

Answer 67

@waterineyes: Thank you so much for all the effort you've put into this.

Answer 68

Any OS user will do the same.. :)

Answer 69

I promise you. The question doesn't have an equal sign.

Answer 70

Yana, there can be omission too, printing mistake can be there..

Answer 71

A function is defined by f(x) = x terms like this, It is its format...

Answer 72

Now let us stick to f(x) = x+9, this we will do for you, and f(x) = -x + 9 this you will do for yourself only.. :P

Answer 73

Lets just stick with this equation first like I have said above:
\(\huge{f(x)=x+9}\)
And my answer is:
\[f ^{-1}(x)=x-9\]

Answer 74

It's Ok @mathmale Not everyone can do it.

Answer 75

Now hurry up..

So, we have with ourselves : \(f^{-1}(x) = x - 9 \)

Answer 76

Would I plug in 5 for x?

Answer 77

Wait, you have f(x) and f^{-1}(x), in which x you are trying to plug in??

Answer 78

x-9

Answer 79

If you want to learn more then:

\[(f \circ f)x \implies f[f(x)]\]

Answer 80

Similarly:
\[(f \circ f^{-1}) \implies f[f^{-1}(x)]\]

Answer 81

So, here you are given with:

\[(f \circ f^{-1})(5) \implies \color{green}{f[f^{-1}(5)]}\]

Answer 82

Ok. So, what would I do next?

Answer 83

You know : f^{-1}(x) = x - 9 and to find f^{-1}(5) you will replace x with 5, so go ahead..

Answer 84

Getting or not??

Answer 85

Yes. Let me type out what I have.

Answer 86

You can write it without LATEX too because I don't want you to start writing Novel again.. :P

Answer 87

Hahhaa!!:D sorry if it really does take that long.
\[f ^{-1}(x)=5-9\]

Answer 88

You can subtract it too otherwise you should have calculator with you if you don't want your brain do this calculation.. :P

Answer 89

And please note since you are replacing x with 5 so replace it on both LHS and RHS..

\(f^{-1}(x) = x - 9 \implies f^{-1}(5) = 5- 9\) like this..

Answer 90

I would plug that into my calculator. Right?

Answer 91

Are you serious??

Answer 92

5 - 9 = ??

Answer 93

Hahha. I was just kidding. I know what is is. It's -4:D

Answer 94

Don't do that otherwise I will tell moderators to investigate how you got to 75.. :P

Answer 95

Omg. Are you for real? What about that \(\huge{\uparrow}\) 90 years old?

Answer 96

\[(f \circ f^{-1})(5) \implies \color{green}{f[f^{-1}(5)]}\] I'd have to check that out; not sure that I'd agree. I'd much prefer you write \[f(f ^{-1}(x))\]FIRST and substitute x=5 LAST.

No, Yana, you do not need a calculator here.


If you start with ANY f)x) and find the inverse of f(x), then the composition of the two separate functions MUST equal x. In fact this is the test for correctness when you are finding the inverse of a function.

Thus, if you are really ending up with f(x) and \[f ^{-1}(x), \] the composition must be and is simply x.

So there we let x=5 and we're done.

Answer 97

So our expression further reduces to:

\[f[f^{-1}(5)] \implies \color{red}{ f[-4]}\]