SOLUTION: Please help with the following as functions is a concept I'm struggling to grasp. if f(t+15) = t^2 - 10, find f(t). I believe this has to deal with tranformations and going i

Question 109093: Please help with the following as functions is a concept I'm struggling to grasp.
if f(t+15) = t^2 - 10, find f(t).
I believe this has to deal with tranformations and going in reverse order.
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Try this as the answer:
.
f(t) = t^2 - 30t + 215
.
Here's the way I worked it:
.
Since f(t+15) has a t^2 term, then f(t) must also have a t^2 term in it so that when you
substitute (t+15) for t you end up with a t^2 plus additional terms.
.
Then if f(t) has contains a t^2 term, when you substitute (t+15) for the t in the t^2 term
you get (t+15)^2 and this squares out to be t^2 + 30t + 225.
.
Since the answer for f(t+15) contains no term with just "t", the +30t must be eliminated.
This tells you that f(t) must have a -30t term so that when you substitute (t+15) into
the -30t term of f(t) you get -30t -30*15 = -30t -450.
.
So far, we've deduced that f(t) consists of t^2 - 30t. If you substitute (t+15) into this
partial deduced form you get:
.
f(t+15) = (t+15)^2 - 30(t+15)
.
and this works out to be:
.
f(t+15) = (t^2 + 30t + 225) - 30t - 450
.
If you combine the like terms (+30t and -30t = 0) and (+225 and -450 = -225) this reduces
to:
.
f(t+15) = t^2 - 225
.
But the problem tells you that f(t+15) should be t^2 - 10. Therefore f(t) must contain
a term +215 so that it combines with the -225 to create the -10
.
So the deduction is that:
.
f(t) = t^2 - 30t + 215
.
To check this out, let's find f(t+15) starting with f(t) = t^2 - 30t + 215
.
Begin by substituting (t+15) for every t in f(t). When you do you get:
.
f(t+15) = (t+15)^2 - 30(t+15) + 215
.
Square out and multiply out the terms on the right side and you get:
.
f(t+15) = t^2 + 30t + 225 - 30t -450 + 215
.
As before, the +30t and the -30t cancel each other out. And now the +225 - 450 and +215
add to -10.
.
Therefore, f(t+15) equals t^2 - 10, just as the problem says it should be.
.
So the answer is that f(t) = t^2 - 30t + 215
.
Hope this helps you to understand the problem. It's hard to explain, and probably is
confusing as well. If you can't understand what I did, repost the problem and maybe another
tutor can do a better job of providing a more understandable process for you to use in
getting the answer.
.
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