To solve this problem, plug into and simplify. Then plug that expression into :

When multiplying an even number of negatives, you get a positive.

When multiplying an odd number of negative, you get a negative.

When multiplying an odd number of negatives, the answer is negative.

When multiplying an even number of negatives, the answer is positive.

.  However,  cannot be simplified any further because the terms have different exponents.

(Like terms are terms that have the same variables with the same exponents. Only like terms can be combined together.)

Change the mixed numbers into improper fractions by multiplying the whole number by the denominator and adding the numerator:

Dividing by a fraction is the same as multiplying by the reciprocal, so the problem becomes .

The domain of a function consists of all of the values of x for which f(x) is defined. When determining the domain of a funciton, the three most important things we want to consider are square roots, logarithms, and denominators of fractions. These tend to signal places where the function is not defined.

First, let's look at the  term. Remember we can only find the square root of nonnegative values. Thus, everything under the square root symbol must be greater than or equal to zero. This tells us that, for this function, .

Second, we need to look at the natural logarithm. The natural logarithm can only be applied to positive numbers (which don't include zero). Thus, everything within the paranethesis of the natural logarithm must be greater than zero.

There are several ways to solve this inequality. One way is to factor the left side and examine the factors. We know that  because of the difference of squares factorization formula. 

.

This statement will only be true in two situations; either both factors must be positive, or both must be negative.

We can see that if , then the factor  will be positive, but the  factor will be negative. If we were to multiply a negative and a positive number, we would get a negative number. Thus,  is not larger than zero when .

Let's consider the interval . In this case, both  and  would be positive. Thus,  when .

Third, consider the interval . In this case, the first factor will be negative, and the second will be positive, so their product would be negative, and  would not be greater than zero. 

To summarize,  only if .

We can see now that f(x) is only defined if  and .

There is one more piece of information we need to consider--the denominator of f(x). Remember that a fraction is not defined if its denominator equals zero. Thus, if the denominator is equal to zero at a certain value of x, we can't include this value of x in the domain of f(x).

We can set the denominator equal to zero and solve to see if there are any values of x where the denominator would be zero.

Rewrite this as an exponential equation. In general, the equation  can be rewritten as , provided that a is positive. 

If we put  into exponential form, we obtain

We can solve this for x.

So, let's put all of this information together. We know that f(x) is only defined if ALL of these conditions are met: 

The only interval for which this is true is if x is greater than (and not equal to) zero but less than (and not equal to) 1. Thus, the domain of f(x) is .

The answer is .

Inequalities can be treated like any other equation except when multiplying and dividing by negative numbers. When multiplying or dividing by negative numbers, we just flip the sign of the inequality so that  becomes , and vice versa.

Inequalities can be treated like any other equation except when multiplying and dividing by negative numbers. When multiplying or dividing by negative numbers, we just flip the sign of the inequality so that  becomes , and vice versa. When we solve binomials, we must take extra caution because .

So when we solve inequalities with binomials, we must create two scenarios: one where the value inside of the parentheses is positive and one where it is negative. For the negative scenario, we must flip the sign as we normally do for inequalities.

Now we must create our two scenarios:

 and 

Notice that in the negative scenario, we flipped the sign of the inequality.

 and 

 and 

Inequalities can be treated like any other equation except when multiplying and dividing by negative numbers. When multiplying or dividing by negative numbers, we just flip the sign of the inequality so that  becomes , and vice versa. When we solve binomials, we must take extra caution because .

So when we solve inequalities with binomials, we must create two scenarios: one where the value inside of the parentheses is positive and one where it is negative. For the negative scenario, we must flip the sign as we normally do for inequalities.

Now we must create our two scenarios:

 and 

Notice that in the negative scenario, we flipped the sign of the inequality.

 and 

 and