Thread: Functions

Let the function f : [3,1) -> R be defined by f(z) = (z −3)^2 +5, and
g : R -> R be defined by g(x) = 7 e^x.
(a) Find ran f.
(b) Find g(ran f).
(c) Let B := ran f and C := g(ran f). Write explicitly the expression for (g)^−1(h) for h E C,
and the expression for (f)^−1(x) for x E B.
(d) Write explicitly the expression for (g o f)^−1(h) for h E C.

Originally Posted by luckyc1423

Let the function f : [3,1) -> R be defined by f(z) = (z −3)^2 +5, and
g : R -> R be defined by g(x) = 7 e^x.
(a) Find ran f.

What is ?
I assume .

The range are all such that for some .
Thus,



To have a solution we need that, .
Thus,

We need that,


We see that cannot work.
Thus,
this tells us that .
It is equivalnet to squaring both sides,


Note it satisfies the condition we have before .

(b) Find g(ran f).

This is defined as,