1. solve the equation x = (2x+ 3)1/2
2. consider f1(x) = ln(x + 1) + ln (x-1) and f2(x) = ln(x^2-1).
a) state domains and ranges of f1 and f2.
b) sketch the curves y = f1(x) and y = f2(x).

1.  ; 2. Domain:    Range: all real numbers

Step-by-step explanation:

Let's find the solutions.

1. Solve the equation so:

Although we have two answers, remember that from the original equation the result of is never negative. So -1 do not solve the equation.

In conlcusion, the equation is solved by x=3.

2A. Domains and ranges of f1(x) and f2(x)

Using logarithmic property we have:

because:

is defined by   then:

so the domain of f1(x) is

Now for the range:

notice that is always positive, so the range of f1(x) is all real numbers.

Be aware that although point number two of the problem mentioned two equations, f1(x)=f2(x) by logarithmic properties, so their domains and ranges are the same.

2B. Graph of f1(x) is attached. Because f1(x)=f2(x) both functions plot equal.

the system of inequity is

step-by-step explanation:

from the given graph it is noticed that we have two related lines. one is solid and another is dashed.

if a line passing through two points, then the equation of line is

the solid line passing through the points (0,4) and (4,0).

the related equation of solid line is .

similarly, find the related equation of dashed line. the dashed liune passing through the points (0,0) and (3,1).

the related equation of dashed line is

from the given graph it is noticed that the point (0,1) lies on the shaded region, therefore the point (0,1) must be satisfied by the related equations.

check the related equations by the point (0,1).

this statement is true if the sign of inequality is < instead of equality. since it is an solid line therefore the sign of inequality will be . the required inequality is

similarly, check the second related equation.

it is a dashed line, so this statement is true if the sign of inequality is > . the required inequality is