The denomiator factors out to:

The denominator becomes zero when . But the function can exist at any other value.

Simplify:

Even though the cancels out from the numerator and denominator, there is still a hole where the function discontinues at . The function also does not exist at , where the denominator becomes .

The square cannot house any negative term or can the denominator be zero. So the lower limit is since cannot be , but any value greater than it is ok. And the upper limit is infinity.

The natural log function does not exist if the inside value is negatuve or zero. The points where the inside becomes negative are or . If is greater than , both terms, and , are positive. If is less than , both terms are negative and multiply to become positive. If the value is between and , only one term will be negative and result in a , which does not exist.

Any value can be inputed in the exponetial.

The value inside a natural log function cannot be negative or . At , the inside is and any value less than cannot be included, because result will be a negative number inside the natural log.

Note that in the denominator, we need to have to make the square root of x defined. In this case is never zero. Hence we have no issue when dividing by this number. Therefore the domain is the set of real numbers that are

Every element of the domain has as image 7.This means that the function is constant . Therefore,

the range of f is :{7}.

Because the only term in the equation containing an is squared, we know that its value will range from (when ) to (as approaches ). When is large, a constant such as does not matter, but when is at its smallest, it does. We can see that when , will be at its minimum of . This number gets bracket notation because there is an value such that .

Exponentials cannot have negatives on the inside. However, the expoential will convert any value into a positive value.