Question

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

f of x equals four divided by x. and g of x equals four divided by x

Answer 1

All i have so far is simply doing as your said and replacing f(x) and g(x) with y and then swapping x and y place

Answer 2

i dont know where to go from here @Hero

Answer 3

Okay, so with any math problem you have, there will always be "Givens". You start with that. In this case, you are given the following:

\(f(x) = \dfrac{4}{x}\) and \(g(x) = \dfrac{4}{x}\). Problem is, when I list \(f(x)\) and \(g(x)\) as you've posted, then are not inverses of each other.

Answer 4

But its telling me that i have to confirm that they are inverses of one another...

Answer 5

not if they are or not.

Answer 6

Where is this problem listed? I would like to see it as originally posted in the book or web source.

Answer 7

Want a screen shot?

Answer 8

Yes! Because I assure you, as you have stated it, they are not inverses. We'll see what the source says.

Answer 9

Confirm that f and g are inverses of one another

Answer 10

Hang on a minute

Answer 11

ok...

Answer 12

Okay, I stand corrected. They are inverses of each other. We have to show why.

Answer 13

So to do that, we simply should evaluate \(f(g(x))\) and \(g(f(x))\).

Answer 14

So how do we do this? We do this by simply replacing the inputs of f and g with the appropriate expressions.

Answer 15

Notice that the input to f is \(g(x)\) and the input to g is \(f(x)\). Keep in mind, we were already given the expressions for both \(f(x)\) and \(g(x)\).

Answer 16

so the input for f is 4/x and the input for g is 4/x?

Answer 17

Correct. Would you mind writing that out using the draw feature?

Answer 18

that ?

Answer 19

Well, actually, you said it correctly, you just did not write it correctly.

Answer 20

oh

Answer 21

If you wrote "The input to f is \(g(x)\)" as a mathematical expression, it would be:\)

\(f(g(x))\). You already know the expression for \(g(x)\) so you would simply replace \(g(x)\) with that given expression.

Answer 22

so the expression for g(x) is 4 over x? sorry im confused

Answer 23

Yes, the expression for \(g(x)\) is \(\dfrac{4}{x}\).

Answer 24

ok so it would be f(4/x) = x?

Answer 25

Well, we're trying to SHOW that actually. But since you've already taken the liberty to do that. We can use it as our original setup like so. Here's what you want to say starting off.
We are given that \(f(x) = \dfrac{4}{x}\) and \(g(x)= \dfrac{4}{x}\). We want to show that

\(f(g(x)) = x\) and \(g(f(x)) = x\)

And then write the replacements immediately below each to show naturally how the steps unfold. I'll show you that next but just for \(f(g(x))\)

Answer 26

ok

Answer 27

I know that looks confusing, but I'll try to explain.

Answer 28

yea lol if you dont mind aha

Answer 29

We will start from where you are stuck. You understand the first two equal signs at least, correct?

Answer 30

yep yep

Answer 31

Great. Also you understand that \(f(x) = \dfrac{x}{4}\) correct?

Answer 32

yea

Answer 33

Okay, so knowing all of that, tell me, if I asked you to evaluate \(f(8)\), would you be able to do so?

Answer 34

If so, show me using the draw feature.

Answer 35

Ah shucks, there's an annoying typo in what I was showing you above.

Answer 36

we would divide f(8) by 8 right?

Answer 37

You know what, I messed up the calculation so bad I have to start over. Let me recalculate then post again. Sorry.

Answer 38

ok lol its ok thanks

Answer 39

\(\begin{align*}
x &=f(g(x))
\\&= f\left( \dfrac{4}{x}\right)
\\&= \dfrac{4}{\dfrac{4}{x}}
\\&=x
\end{align*}\)

Answer 40

There we go now it's fixed.

Answer 41

ok

Answer 42

So in order to figure out how to do these, we must know how to do basic calculations such as evaluating the function at a particular value.

Answer 43

So if you were asked to evaluate \(f(8)\) you would do the following:

Write out the original function:
\(f(x) = \dfrac{4}{x}\)

Then replace x with 8 everywhere you see x:
\(f(8) = \dfrac{4}{8}\)

Then simplify the left side of the equation:
\(f(8) = \dfrac{1}{2}\)

Answer 44

Take some time to think about what I just did, then evaluate \(f(2)\).

Answer 45

f(x) = 4/x
f(2) = 4/2
f(2) = 2?

Answer 46

Correct. Now you have it. Now evaluate f(1/2)

Answer 47

f(x) = 4/x
f(1/2) = 4/.5
f(1/2) = 8?

Answer 48

Yes, correct. So now, that you're getting the hang of it. What is f(x/2)?

Answer 49

for the one we just did?

Answer 50

Please do not get confused on me now bro. I showed you how to do f(8) and then you did f(2) and f(1/2) on your own with no problem. So now do f(x/2).

Answer 51

You follow the same procedure as you did for the previous evaluations.

Answer 52

i know but i had a number i get confused when the letters are thrown in...

Answer 53

Please try your best. Use the draw feature this time.

Answer 54

this is where im stumped i dont know how to divide it by x/2

Answer 55

\(f\left(\dfrac{x}{2}\right) = \dfrac{4}{\dfrac{x}{2}} = 4 \div \dfrac{x}{2} = 4 \times \dfrac{2}{x}\)

Answer 56

Is that a good enough hint for you?

Answer 57

@SourMunchkin7806 are you still alive?