Function Machines: 6-12, 8-20 | Free Homework Help

[Natasha1]

Homework Statement

There are two possible functions machines which will give the answers below:
6 --> ? then ? --> 12
8 --> ? then ? --> 20

Homework Equations

[/B]
I have managed to find the first which is x4 and then - 12 so:
6 --> x4 then -12 --> 12
8 --> x4 then -12 --> 20 works!

The Attempt at a Solution

I can't find the second... Any helps please...

 

[BvU]

You sure ? 6 --> ? would be 6 --> 12 but then ? --> 12 no longer is satisfied !
In other words: you are looking for some function f that satisfies

f(6) = a --> f(a) = 12 and
f(8) = b --> f(b) = 20
​

Sorry to spoil the fun ...

 

[SammyS]

Homework Statement

There are two possible functions machines which will give the answers below:
6 --> ? then ? --> 12
8 --> ? then ? --> 20

Homework Equations

[/B]
I have managed to find the first which is x4 and then - 12 so:
6 --> x4 then -12 --> 12
8 --> x4 then -12 --> 20 works!

The Attempt at a Solution

I can't find the second... Any helps please...

I take it you mean that ƒ(x) = 4x - 12.

As BvU points out you need to apply your function twice to 6, resulting in 12, and twice to 8 to get 20.

Therefore, you need to compose function, ƒ, with itself to get x → 4x - 12 .

 

[Samy_A]

You could write your function machine as $f(x)=ax+b$, where $a$ and $b$ are to be determined.

As @SammyS said, you have to apply the function twice to 6 to get 12, and twice to 8 to get 20.

For example:
$f(6)=6a+b$,
$f(f(6))=f(6a+b)=a(6a+b)+b=6a²+ab+b=12$

Do the same for $f(f(8))$, then you will have two equations that can be solved for $a$ and $b$.

 

[Natasha1]

got it, it is -3 and then x4

 

[HallsofIvy]

This is the same answer as before: f(x)= 4(x- 3)= 4x- 12. This has the property that f(6)= 12 and f(8)= 20 but that is NOT how I would interpret the question- and not how the others responding interpret it. We are reading it as asking for two different functions, f and g, such that f(f(6))= 12, f(f(8))= 20, g(g(6))= 12. and g(g(8))= 20! But perhaps we are interpreting this incorrectly. You are certainly using unusual notation- it is far more common to write "4x" rather than "x4" to indicate "4 times x". Was that what you were taught in class? And when you say "two functions" do you mean what I would call two parts of one function? That is are you thinking of "4 times x" and "subtract 12" as the "two functions"?

 

[mfb]

I think the question is meant as g(f(6))=12, g(f(8))=20, find g and f. There are many possible pairs of functions, the two found by Natasha1 are the easiest ones (and the only pairs that involve just a single elementary operation per "machine").

 

[Samy_A]

I think the question is meant as g(f(6))=12, g(f(8))=20, find g and f. There are many possible pairs of functions, the two found by Natasha1 are the easiest ones (and the only pairs that involve just a single elementary operation per "machine").

Yes, that may be the correct interpretation of the question. As others I thought it meant f(f(6))=12 ...

 

[SammyS]

I think the question is meant as g(f(6))=12, g(f(8))=20, find g and f. There are many possible pairs of functions, the two found by Natasha1 are the easiest ones (and the only pairs that involve just a single elementary operation per "machine").

(I'm pretty sure that Natasha1 has found only one such composite machine. After all, 4x − 12 = (x − 3)⋅4 , as pointed out by Halls.)

I doubt that this is the correct interpretation.

As you say, if we are looking for functions, ƒ and g, such that ƒ○g maps 6 → 12 and 8 → 20, there are many such pairs. I suspect that there are infinitely many such pairs.

OP has discovered a linear function, let's call it h, such that h(6) = 12 and h(8) = 20. That function is defined by h(x) = 4x − 12.

A nice characteristic of linear functions is that the composition of two linear functions is a linear function.

There are two linear functions, ƒ and g such that ƒ○ƒ = g○g = h , where h(x) = 4x − 12, i.e. ƒ(ƒ(x)) = g(g(x)) = 4x − 12 . I am convinced that these are the two functions intended by the author of the problem.

 

[mfb]

The language of the problem statement is very basic, and other problems are on a level similar to the interpretation I posted.

 

[SammyS]

The language of the problem statement is very basic, and other problems are on a level similar to the interpretation I posted.

I think I finally see your point.

Let ƒ(x) = 4x and g (x) = x − 12 .

Then g (ƒ(x)) = 4x − 12 .

Similarly, ƒ(x) = x − 3 and g (x) = 4x gives g (ƒ(x)) = 4x − 12 .

The first pair seems to be what OP means in post #1, by saying "x4", meaning multiply by 4, and then "− 12" meaning → take that result and subtract 12.

In post #5, OP describes the second pair in the same manner.

 

[symbolipoint]

This does not need to be made over-complicated. If you just have two input-output pair, then maybe a linear function may be good enough. What line contains the points, (6,12) and (8,20)? Put the equation into slope-intercept form, and name your function.

 

[Ray Vickson]

I think the question is meant as g(f(6))=12, g(f(8))=20, find g and f. There are many possible pairs of functions, the two found by Natasha1 are the easiest ones (and the only pairs that involve just a single elementary operation per "machine").

The wording is ambiguous, but to me it seems to read as: "there are two functions $f = f_1$ and $f = f_2$ that give $f(f(6)) = 12, f(f(8)) = 20$". Of course, there are (possibly) infinitely many such functions, but if we restrict attention to linear functions of the form $f(x) = a + bx$ then, indeed, there are exactly two of them that "work".

 

[Natasha1]

Thank you everyone, I have understood

 

[symbolipoint]

Having only two data points allows for a linear relation, USUALLY, depending on the situation given or the situation expected.