Question

Let f(x) = 4x − 3. If f(a) = 9 and f(b) = 5, then what is f(a + b) ?

Answer 1

What you are going to want to do with this problem is add up

(4(9) -3) + (4(5) - 3)

Do you understand how I substituted those values?

Answer 2

No I don't, if you don't mind could you explain it?

Answer 3

Because the problem requires f(a) + f(b), I added up the equation with the value of f(a) and f(b) together.

f(a) = 4(9) - 3
f(b) = 4(5) - 3

I simply added the two equations together.
I got the values from the equations given.

Answer 4

no, it requires \(f(a+b)\ which is not necessarily the same as \(f(a)+f(b)\!

Answer 5

True whpalmer, so regard that note. Besides the mistake the equation is still set up correctly. I hope that doesn't confuse you too much.

Answer 6

no kirby f(a)= 9 means 4*a - 3 = 9 making a = 3

Answer 7

if \(f(x) = 4x-3\) and \(f(a) = 9\) then
\[f(a) = 4a-3 = 9\]solving for \(a\)
\[4a-3=9\]\[4a=12\]\[a=3\]
Now do the same for \(b\), then add \(a+b\) and plug it into \(f(x) = 4x-3\)

Answer 8

b = 2

so we need to find f(2+3) = f(5)

Answer 9

Thank you.

Answer 10

Notice that \(x\mapsto4x-3\) is an affine map so we can see that \(f(a+b)=f(a)+f(b)+3\) ergo \(f(a+b)=9+5+3=17\)

Answer 11

no need to even solve for \(a,b\) just reason it out...$$f(a+b)=4(a+b)-3=(4a-3)+(4b-3)+3=f(a)+f(b)+3$$

Answer 12

true enough, but someone asking for help with this problem is unlikely to know about affine maps, and likely to draw the wrong conclusion that it is safe to assume that f(a+b) = f(a) + f(b) in the future

Answer 13

nobody said \(f(a+b)=f(a)+f(b)\) but if \(f(x)=kx\) it's rather obvious that \(f(a+b)=f(a)+f(b)\)... so naturally if we shift \(f\) vertically by \(h\) you can intuitively see why \(f(a+b)-h=f(a)-h+f(b)-h\) ergo \(f(a+b)=f(a)+f(b)-h\)

Answer 14

actually, if you re-read some of the first responses, you'll see that someone did.

Answer 15

alright well I meant in this current conversation... I pay no attention to wrong answers

Answer 16

:-p

Answer 17

if only we could assume the original poster had the same filter, our life would be much easier :-)

Answer 18

anyhow, thanks for providing me with the "learn something new every day" moment (hadn't heard of affine maps) :-)