Finding the formula of a function based on output

Question

This is probably something super simple, but I can't find it in my book, and I don't even know what to search for because I don't know what to call it.

I'm not looking for this specific answer, but how would I approach a problem like this.

$$ \mbox{If}\ f(x - 4) = 3x + 2,\ \mbox{find}\ f(-2).$$

So, $3x+2$ is the output, being modified by the formula I need to find. I don't know how to algebraically find that equation without brute forcing. For what it's worth, the answer above was apparently $-12$.

Edit. Answer was not -12, but still had trouble figuring out how to approach to find correct answer.

2 would be the x value there, but I'm trying to find what f(x), knowing that if I pass in x-4, it'll spit out 3x+4 — Robert, Jan 23 '15 at 07:31
I've tried asking some friends, and they seem to approach it as f(x)=3x+4, and start working with 3x+4 to find it, in a way such as 3(-2 - 4) + 4, but that outputs -14, not the expected -12 — Robert, Jan 23 '15 at 07:33
If you prefer, let $y=x-4$, then $x=y+4$ and $f(y) = 3(y+4)+4 = 3y + 16$. Then $f(-2) = 3(-2)+16 = 10$. — copper.hat, Jan 23 '15 at 07:36
So, essentially, are you treating f(x-4)=3x+4 as a whole new formula, and substituting -2 in for x? — Robert, Jan 23 '15 at 07:39
Let us continue this discussion in chat. (it popped up a link saying this...?) — Robert, Jan 23 '15 at 07:41
I don't know what your last comment means. The function $f$ is defined by the pairs of numbers $(x-4,3x+\text{whatever})$. So, you want to find the right value when the left value is $-2$. — copper.hat, Jan 23 '15 at 07:41
What would I call this sort of operation so I can research it further, I don't even know what to look up. — Robert, Jan 23 '15 at 07:43

k170 · Accepted Answer · 2015-01-23T16:34:59.900

1

Here are the steps in full detail $$ f(x-4)=3x+2 $$ $$ \frac13 f(x-4)=x+\frac23 $$ $$ \frac13 f(x-4)=x+\frac23 +\frac13-\frac13 $$ $$ \frac13 f(x-4)=x+1-\frac13 $$ $$ \frac13 f(x-4)=x+1-5+5-\frac13 $$ $$ \frac13 f(x-4)=x-4+5-\frac13 $$ $$ f(x-4)=3(x-4)+15-1 $$ $$ f(x-4)=3(x-4)+14 $$ $$ f(s)=3s+14 $$ So now we have $$ f(-2)=3(-2)+14 =-6+14=8$$ We can also skip all of this and just use the fact that $$ x-4 =-2 $$ $$ x =-2 +4 =2 $$ So now we have $$ f(2-4)=f(-2)=3(2)+2 $$ $$=6+2=8 $$ Either way, $$f(-2)\not =-12$$

edited Jan 23 '15 at 16:34

answered Jan 23 '15 at 16:26

k170

9,045

I think it should be $f(s) = 3s+16$. – copper.hat Jan 23 '15 at 16:43
@copper.hat, hmm where do you suppose the error is in my answer? – k170 Jan 23 '15 at 16:53
I just realised that there are two different formulas for $f$ in the question. They differ by 2. – copper.hat Jan 23 '15 at 16:55
So I stayed up most of the night messing with this formula, and the way that seemed closest was x-4=-2, then x=2, and you can use that to solve in the 3x+2. – Robert Jan 24 '15 at 23:13

Laars Helenius · Answer 2 · 2015-01-23T07:39:26.710

0

If you want $f(-2)$ and you know that $f(x-4)=3x+2$, then you want $x-4=-2$. Does that help?

By the way, I think your supposed answer of $-12$ is incorrect.

edited Jan 23 '15 at 07:39

answered Jan 23 '15 at 07:34

Laars Helenius

7,965
1
23
35

The formula has changed. – copper.hat Jan 23 '15 at 07:36
Got it. My phone app doesn't update quickly. – Laars Helenius Jan 23 '15 at 07:42

score 0 · Answer 3 · answered Jan 23 '15 at 18:09

There are two ways to look at your question.

1) you just want to compute $f(-2)$. For this, you need to find $x$ such that $x-4=-2$. Obviously, $x=2$ and $f(-2)=3\cdot2+2$.

$$f(-2)=8.$$

2) you want to get the expression for $f(x)$ instead of $f(x-4)$. In this case, use a change of variable $y=x-4$, hence $x=y+4$, and $f(y)=3x+2=3y+14$.

$$f(x)=3x+14.$$

Finding the formula of a function based on output

3 Answers3