$\"\"$

(a)

Yes, the function $\"\"$ is exist.

Because the each population of U.S. corresponds to exactly one particular year.

So the function P is one - to - one function. \ \

Thus, the function $\"\"$ is exist.

$\"\"$

(b)

The $\"\"$ represents the time in years for a given population level. $\"\"$

(c)

Observe the given table of values :

The value 357.5 corresponds to exactly one value, i.e, 25.

Thus, the value of $\"\"$ .

$\"\"$

(d)

No, $\"\"$ is not exist at 2050.

Because then the population for 2030 and 2050 would be the same(The function values $\"\"$ and $\"\"$ ).

Thus, the function is not a one - to - one.

So, there is no possibility for inverse.

$\"\"$

(a) Yes.

(b) The $\"\"$ represents the time in years for a given population level.

(d) No.