Domain and Range - Pre-Calculus

We know that . So .

Therefore:

.

This gives:

.

Therefore the range is:

Note that: we can write f(x) as :

.

Since,

Therefore,

So the range is

We have .

Adding 7 to both sides we have:

.

Therefore .

This means that the range of f is

The part inside the square root must be positive. This means that we must be . Thus

. Adding -121 to both sides gives . Finally multiplying both sides by (-1) give:

with x reals. This gives the answer.

Note: When we divide by a negative we need to flip our sign.

cos(x) is definded for all reals. cos(x) is always between -1 and 1. Thus . The value inside the square is always positive. Therefore the domain is the set of all real numbers.

Since

for all real numbers, the denominator is never 0 .Therefore the domain is the set

of all real numbers.

Apart from the multiplication by , this function is very similar to the function

The logarithmic function has domain x>0, meaning for every value x>0, the function has an output (and for x = 0 or below, there are no values for log x)

The range indicates all the values that can be outputs of the . When you draw the graph of y = log(x), you can see that the function extends from -infinity (near x = 0), and then extends out infinitely in the positive x direction.

Therefore the Domain:

and the is Range:

The domain is the set of x-values for which the function is defined.

The range is the set of y-values for which the function is defined.

Because the values for x can be any number in the reals,

and the values for y are never negative,

Domain: All real numbers

Range:

The domain is the set of x-values for which the function is defined.

The range is the set of y-values for which the function is defined.

Because the values for x are never negative,

and the values for y are never negative,

Domain:

Range:

As long as the number under the square root sign is greater than or equal to , then the corresponding x-value is in the domain. So to figure out our domain, it is easiest to look at the equation and determine what is NOT in the domain. We do this by solving and we get . We now look at values greater than and less than , and we can see that when , the number under the square root will be negative. When , the number will be greater than or equal to . Therefore, our domain is anything greater than or equal to 6, or .

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 .

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 denominator becomes when or , so the function does not exist at these points. In numerator, must be at least or greater to be real. So the function is continuous from to and to any other value greater than .

The denomiator factors out to:

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

Cannot have a negative inside the square root. The value of has to be for the inside of the square root to be at least . This is the lower bound of the domain. Any value of greater than exists.

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.

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