Incomplete function table... with no rule. Not sure what to do here.

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mickey222

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I tried a lot of things... the first rudimentary thing I did was attempt find the difference between some of the input values, and then the difference between some of the output values. Didn't work. Then I tried plugging them into a slope formula... I feel like an idiot. Not sure what to do.

Oh and the ones I got right, I guessed.
 
Let u = f(3) and v = g(3)
(f-g)(3) = f(3) - g(3) = -12 --> u - v = -12
(f+g)(3) = f(3) + g(3) = 4 --> u + v = 4

Now solve for u and v, and you'll have f(3) and g(3). Repeat that logic for other missing values.
 
mickey222 said:
View attachment 22451


I tried a lot of things... the first rudimentary thing I did was attempt find the difference between some of the input values, and then the difference between some of the output values. Didn't work. Then I tried plugging them into a slope formula... I feel like an idiot. Not sure what to do.

Oh and the ones I got right, I guessed.
Click to expand...
It took me a while to realize what this is asking you to do, because of your title. It is not about guessing a rule! You don't even need to look at the values of x, really. Nor should you look at one row at a time, as you seem to be doing. Instead, look at one column at a time.

In each column, you are given values of two or three of f, g, f-g, and f+g, and are to find the missing number(s) in that column.

In the first column, you are told that f=-5 and g=3. They correctly tell you that f+g=-2; this is probably meant to help you see what it means. So what is f-g?

Things get harder as you move to the right.

In the last column, you know that f=-4, f-g=-12, and f+g=4. Since f=-4, you know that -4-g=-12 and -4+g=4, and you can solve for f and g.
 
The motivation for this exercise is to get you to understand what a particular notation means.

[MATH](f + g)(x) \equiv f(x) + g(x) \text { and } (f - g)(x) \equiv f(x) - g(x)[/MATH].

If we let y = f(x) and z = g(x), then (f + g)(x) = y + z and (f - g)(x) = y - z.

So the first box to fill in requires EVALUATING y - z when y = -5 and z = 3.

Every single one of the boxes is at about that level of arithmetic complexity.

The whole point is to get you to see what (f + g)(x) and (f - g)(x) mean.
 
Sorry guys, I've read all your responses and tried to imitate and understand your work. I'm able to work out some of the answers but not all of them.

The first two rows I still don't understand how to solve. How am I suppose to find f(x) and g(x)? For example, in the last column, could someone explain to me how f(3)=-4?

This was how I attempted to solve it:

f(3)=u
g(3)=v

u-v=-12
therefore u=-12+v
therefore f(3)=g(3)-12

Knowing neither f(3) nor g(3), I got stuck here.
 
Last edited:
R.M. said:
Let u = f(3) and v = g(3)
(f-g)(3) = f(3) - g(3) = -12 --> u - v = -12
(f+g)(3) = f(3) + g(3) = 4 --> u + v = 4

Now solve for u and v, and you'll have f(3) and g(3). Repeat that logic for other missing values.
Click to expand...

What do you mean solve for U and V? I did and got u=4-v. Is that what you mean?
 
mickey222 said:
What do you mean solve for U and V? I did and got u=4-v. Is that what you mean?
Click to expand...
Yes - now use that in:

u - v = -12  →  \displaystyle \ \to \ \ (4 - v) - v = -12 ....... now solve for 'v' and then solve for 'u'
 
mickey222 said:
Sorry guys, I've read all your responses and tried to imitate and understand your work. I'm able to work out some of the answers but not all of them.

The first two rows I still don't understand how to solve. How am I suppose to find f(x) and g(x)? For example, in the last column, could someone explain to me how f(3)=-4?

This was how I attempted to solve it:

f(3)=u
g(3)=v

u-v=-12
therefore u=-12+v
therefore f(3)=g(3)-12

Knowing neither f(3) nor g(3), I got stuck here.
Click to expand...
Please don't think in terms of rows. Just look at one column at a time, starting at the left where it is easiest.

Have you worked out the first two columns? Please show us your answers. Then do the next two, where there are two blanks, but you can find one and then the other.

The last column requires that you know how to solve a system of two equations, namely, as has been said, u - v = -12, u + v = 4. Have you ever done that before? You are trying the substitution method; there is also the addition method, which is particularly easy in this case.
 
mickey222 said:
Sorry guys, I've read all your responses and tried to imitate and understand your work. I'm able to work out some of the answers but not all of them.

The first two rows I still don't understand how to solve. How am I suppose to find f(x) and g(x)? For example, in the last column, could someone explain to me how f(3)=-4?
Click to expand...
You are told that f(3) = 4. There is nothing to figure out about f(3).

You are being confused by the idiotic problems that ask you to guess the next number in the row. Please reread my previous post: this is very simple exercise once you understand what it is asking.
 
JeffM said:
You are told that f(3) = 4. There is nothing to figure out about f(3).
Click to expand...
No, f(3) was filled in (correctly) with -3, but was not given. It can be worked out without being told it.

We are given only that (f-g)(3) = -12 and (f+g)(3) = 4. So if f(3) = u and g(3) = v, we know that u-v = -12 and u+v = 4.
 
Dr.Peterson said:
No, f(3) was filled in (correctly) with -3, but was not given. It can be worked out without being told it.

We are given only that (f-g)(3) = -12 and (f+g)(3) = 4. So if f(3) = u and g(3) = v, we know that u-v = -12 and u+v = 4.
Click to expand...
Sorry about that. The number was present in the OP. I failed to notice that it was boxed.

The point, on which you and I agree, is that given two numbers in a column the other two are determinable by simple evaluation.

This could be good exercise if the OP’s mind had not been scrambled by those inane exercises with no correct answers.
 
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