**Functions - Domain and Range; Composition**

**Sol 1** f is defined for all values of x (since f is a polynomial), so the
domain of f is
. Since the graph of f is a parabola which
opens downward with vertex at , the set of y-coordinates for the points
on the graph of f consists of all y-values with ; so the range of f
is the interval .

**Sol 2** f is defined where or , so the domain of f is the
interval . Since
,
for
any x in the domain of f; so the range of f is contained in .
If ,
(since ), so the range of f is actually equal to .

**Sol 3** f is defined for or , so the domain of f is
given by
. To find the range of f, we must determine for
which y-values the equation
has a solution for x.
Multiplying both sides of this equation by gives or
, so
.
Therefore the equation
has a solution for x iff
or , so the range of f is given by
.

**Sol 4** f is defined wherever (so the square root is defined)
and (so the fraction is defined). Solving the
inequality or gives , so the
domain of f is .

**Sol 5**
, while
.

**Sol 6** We can let and , for example.

**Sol 7** Since
,
and therefore
and .

**Sol 8** is defined where , so gives
. Taking the nonnegative square root of both sides gives
or . Therefore is the domain of .

**Sol 9** is defined where

, so factoring gives the inequality

.

Marking 0,5,3, and -3 on a number line and using the facts that all factors have odd exponents and that ,

we get the following sign chart for :

Therefore the domain of is given by

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