|
Question 252190: Suppose f and g are linear functions such that f (g(t )) = t for all real t .
Find the rule for f (x) if f (5) = 3 and g(2) = 3 .
Answer by drk(1908)   (Show Source):
You can put this solution on YOUR website! This may be a longer way than what you want . . .
Let
(i) F(t) = at + b
(ii) G(t) = ct + d
F(G(t)) = act + ad + b = t
We know two things from this: (1) ad = -b ; (2) a = 1/c.
By substitution, into (i) and (ii), we get
(iii) F(t) = (1/c)t - (1/c)d
(iv) G(t) = ct + d
substituting our coordinates, we get
(v) 3 = (1/c)*5 - d/c --> 3c = 5 - d
(vi) 3 = 2c + d --> 3 = 2c + d
adding these together, we get
3c + 3 = 5 + 2c. Solving for c, we get c = 2. Then from (vi), d = -1.
Now from (iii), we have F(t) = (1/2)t + (1/2).
|
|
|
| |
