Math-Please help...explain how you got the answer.?

Question

HIV Infection: The time interval between a person’s initial infection with HIV and that person’s eventual development of AIDS symptoms is an important issue. The method of infection with HIV affects the time interval before AIDS develops. One study of HIV patients who were infected by intravenous drug use found that 17% of the patients had AIDS after 4 years, and 33% had developed the disease after 7 years. The relationship between the time interval and the percentage of patients with AIDS can be modeled accurately with a linear equation.

a.)Write a linear function y=mx+b that models this data, using ordered pairs (4, 0.17) and (7, 0.33).

b.)Use your equation from part a to predict the number of years before half of these patients will have AIDS.

Find domain of the functions:

2.) f(x)=(√16-x2) --square root

3.) f(x)=| x-1|

4.) f(x)=(2) / (x2-4)

Answer 1

1.

a).

y = mx + b

You are given:

0.17 = 4m + b

0.33 = 7m + b

Subtract second from first:

0.16 = 3m

Hence, m = 0.16/3 = 0.05333333333333...

Substitute this in first equation to get b.

0.17 = 4(0.05333333333) + b

b = 0.17 - 0.2133333333333 = -0.043333333333.......

Linear function: y = 0.05333333333x - 0.04333333333

b).

0.5 = 0.05333333333x - 0.04333333333

Solve for x (years):

0.053333333333x = 0.54333333333

x = 0.54333333333/0.053333333333 = 10.1875 years

2).

f(x) will be defined if 16 - x² >= 0

==> x² <= 16 ==> -4 <= x <= 4

Domain of f: [-4, 4]

3).

f(x) defined for all values of x.

Domain of f: R

4).

f(x) defined for all x such that x² - 4 ≠0,

i.e.   x² ≠4

==> x ≠±2

Domain of f: R, excluding -2 and +2