The following formula models the number of prison inmates, p, in thousands, who are in u. s. prisons x years after 1980.

p
=
2
x
2
+
22
x
+
322
p=2x2+22x+322

find the first year after 1980 in which the population of u. s. prisons was approximately 574 thousand people.

1987

Step-by-step explanation:

If we consider x years after 1980, when the number of prisoners, P, in thousands, is given by  

P = 2x² + 22x + 322 (1)

Now, If the population of U.S. prisoners was approximately 574 thousand people, then we have to find the year of this situation.

Now, from equation (1), we get

574 = 2x² + 22x + 322

⇒ 2x² + 22x - 252 = 0

⇒ x² + 11x - 126 = 0

⇒ x² + 18x - 7x - 126 = 0

⇒ (x + 18) ( x - 7) = 0

⇒ x = 7  

As x can not be negative so we neglect the x = - 18 value.

So, in the year 1987, the number of U.S. prisoners became approximately 574 thousand people. (Answer)

guess at question:

a vertical line intersects the graph of g(x,y) = 0 at two points. what conclusion follows about the function f(x) implicitly defined by g(x,f(x)) = 0?

the domain of f(x) cannot include the x coordinate of any such vertical line.

informally, f(x) is not a function, and

g(x,y)=0 cannot be solved for y uniquely.

the relation r = {(x,y) such that g(x,y) = 0} is a function if

for all (a,b) ∊ r and (c,d) ∊ r, a=c implies b=d.

step-by-step explanation: