jj_w

I have looked at this question a few times, and still haven't figured out the solution...

" |x | f(x)|
________
|0 | a |
________
|1 | 24|
_________
2 | b |

The table above shows some values for the function f. If f is a linear function, what is the value of a + b?

(A) 24
(B) 36
(C) 48
(D) 72
(E) It cannot be determined by the given information.

----------------------------------------------------------------------
[sorry, my table-making skills are limited]

Any help would be appreciated. Thanks

mathwiz

what question?

KRabble88

Wait a second.. it's definitely C!

You have your points (0, a), (1, 24), (2, b)

Since this is a linear function, these points will lie all on the same line, therefore the slope between each point will be equal to one another.

So..

(24-a)/(1-0) = (24-b)/(1-2)

24-a = -24 + b

48 = a + b

I hope that's clear

mathwiz

yup, a=0 because f is linear. f(2) = 2f(1) = f*24 = 48. So a+b = 0 + 48 =48. (C)

KRabble88

mathwiz, how do you assume that the line passes through the origin? Can't a function still be linear but have a non-zero y-intercept? I thought a function is linear because it is dependent upon slope not y-intercept. Correct me if I'm wrong on this

mathwiz

Nope. A function is linear if f(cx) = c(f(x)) and f(a+b) = f(a) + f(b). The ones you are thinking of are straight lines.

KRabble88

I need to learn the definitions of some of these math terms. A linear function doesn't necessarily have to be straight? Geez! Why do they make everything so confusing lol..

mathwiz

No, a linear function has to be straight. But there are lots of "straight" functions that aren't linear, such as f=4x+4

jj_w

Thanks! I understand it now!

KRabble88

O okay, thanks mathwiz. So basically:

f(1) = 24

f(1 + 0) = 24 + a (so 'a' is zero)

f(1 + 1) = 24 + b

so, f(2) = f(1) + f(1)

and f(1 + 1) = 24 + 24

f(2) = 48 or f(2) = b

and 'b' will be 48

Does a linear function always have to pass through the origin?

optimizerdad

I personally like Krabble's explanation in post#3, it's clear and simple. A messier way is to do it via straight algebra:
Suppose f(x) = mx + c
Then f(0) = a = (m)(0) + c ; from which c=a
f(1) = (m)(1) + a = 24 ; from which m = 24-a
f(2) = (m)(2) + a = b
or (2)(24-a) + a = b
48 -2a + a = b
48 = a + b

youknowme

why doesn't f=4x+4 a linear function?

optimizerdad

See http://www.answers.com/topic/linear-equation

Strictly speaking, f(x) = mx + c is a linear equation or a linear function.

A function that also satisfies
f(x + y) = f(x) + f(y)
and f(ax) = af(x)
would be a 'linear operator', if my fading memory is correct.

For SAT I and SAT II questions, I'm pretty sure they mean f(x)=mx + c when they talk about linear functions.

gcf101

Points A(0,a), B(1,24) and C(2,b) lie on a straight line.
Point B is exactly in the middle between points A and C (notice the order of x-coordinates of these points), so
24 is exactly in the middle between a and b on the number line, i.e.
24 is the mean of a and b
which means a + b = 2*24,
a + b = 48. That's (C).

PS.
Values of a and b CANNOT be determined from the information given.
That makes choice (E) tempting.
But: if SAT asks for the value of an expression with more than one variable, you almost always can't or needn't find the value of each variable separately.

In this question a and b could be any numbers as long as they add to 48.
Three points (0,0), (1,24), (2,48) on the straight line are as good as
(0,24), (1,24), (2,24), or
(0,48), (1,24), (2,0).

gcf101

PPS.
See a school definition of a linear function
http://www.answers.com/main/ntquery?...&curtab=2222_1
vs. mathwiz's true math definition
http://www.answers.com/topic/linear
(scroll down to Mathematics).

And they say math is a precise science...