how to find $f(2)$

Question

Hello im trying to solve a homework problem and i'm stumped. here is the problem.

the question states suppose $f$ and $g$ are continuous functions such that $g(2)=6$ and $\lim_{x\to 2}$ $[3f(x)+f(x)g(x)]=36$ find $f(2)$

i have tried solving it by doing the following but i get the wrong answer.

3$f(2)$+$f(2)$g(2)=36
3$f(2)$ +$f$(2)6=36
$f(2)$+$f(2)$= 36-18
$f(2)$ +$f(2)$ = 16

how do i finish the problem? the answer is 4 but i dont see how to get there now? Thanks for any help in advance. Thanks Miguel

What happened on the third line? Then what happened on the fourth line? — Jonas Meyer, Feb 02 '13 at 07:19
i took the 3 from the far left and i multiply times 6 to get 18 and subtract from the other side. — Miguel, Feb 02 '13 at 07:20
Typically $ab+bc\neq ab+b+ac$, but rather, $ab+bc=(a+c)b$. For example, $3\times 12 +12\times 6 \neq 3\times 12 + 12 +18$, but rather, $3\times 12+12\times 6 =9\times 12$. — Jonas Meyer, Feb 02 '13 at 07:23
The recent edit replaced one error with another, implying that $36-18=16$. — Jonas Meyer, Feb 02 '13 at 07:25
Jonas thank you this is a very silly mistake on my part i totally see my error now. thank you and i appreciate your respond. — Miguel, Feb 02 '13 at 07:30
You're welcome. I appreciate your question because of its clarity and explanation. — Jonas Meyer, Feb 02 '13 at 07:32

score 2 · Accepted Answer · edited Feb 02 '13 at 07:31

2

You got it all right up until the third line,

$3f(2)+f(2)g(2)=36$

$3f(2) +f(2)6=36$

$3f(2)+f(2)= 36-18$ <- not sure what you are doing here, but it doesn't look right

Anyway, continuing from the second line,

$3f(2) +f(2)6=36$

$f(2)(3+6)=36$

$f(2) = 36/9 = 4$

edited Feb 02 '13 at 07:31

Vikram

answered Feb 02 '13 at 07:18

thats where i get confused... i have 2f(2) so how do i end up with f(2)(3+6)=36? – Miguel Feb 02 '13 at 07:22
sorry, what did you mean by 2f(2)? After, 3f(2)+6f(2)=36, all I did was to factor out f(2), which gives f(2)(3+6) = 36 – Feb 02 '13 at 07:25
ahhhhh !!!! factor thats where im going wrong. im such an idiot. thank you . – Miguel Feb 02 '13 at 07:29
Thank you for the response. i get it now. – Miguel Feb 02 '13 at 07:31

score 1 · Answer 2 · answered Feb 02 '13 at 07:22

1

From $$3f(2)+f(2)6=36,$$ we have $$9f(2)=36 \Rightarrow f(2)=4$$

answered Feb 02 '13 at 07:22

M. Strochyk

what happens two the second f(2)? thats where im stumped. how can you end up with 9f(2)=36 when their are two 2f(2)'s? – Miguel Feb 02 '13 at 07:25
@Miguel: The distributive property of multiplication and addition says that $ab + ac = a(b+c)$. For example, $3\times 4+3\times 26 = 3\times (4+26)$, $6x+82x = 88x$, $3f(2)+f(2)6=3f(2)+6f(2)=9f(2)$, etc. Or if you prefer, "factor out" $f(2)$. – Jonas Meyer Feb 02 '13 at 07:28
It means that $f(2)$ is multiplied by $g(2)=6$: $$f(2)g(2)=f(2)\cdot6=6\cdot f(2).$$ Next by collecting terms $3f(2)$ and $6f(2)$ we have $3f(2)+6f(2)=(3+6)f(2)=9f(2)$ – M. Strochyk Feb 02 '13 at 07:34
$3f(2) +f(2)6=36$ means finding the value of function at 2 and then multiplying that value by 3, same way finding the value of function at 2 and then multiplying that value by 6 and then add both to get 36, so you have finding the value of function at 2 and then multiplying that value by $(3+6)$ – Vikram Feb 02 '13 at 07:36

2 Answers2