Relations and Functions - Pre-Calculus

The relation is not a function because and hold. If it were a function, would hold only for one . But we know it holds for because and . Thus, the relation on the set of real numbers is not a function.

The statement

"Even if the two parents had only one daughter, the relation would not be a function."

is not true because if they had only one daughter, say Adriana, then the only relations that would exist would be (Juan, Adriana) and (Oksana,Adriana), which defines a function.

The definition of a function requires that for each input (i.e. each value of ), there is only one output (i.e. one value of ). For , each value of corresponds to two values of (for example, when , both and are correct solutions). Therefore, this relation cannot be a function.

A relation is a function if no single x-value corresponds to more than one y-value.

Because the mapping from goes to and

the relation is NOT a function.

First find the reciprocal of the slope of the given function.

The perpendicular function is:

Now we must find the constant, , by using the given point that the perpendicular crosses.

solve for :

A set of ordered pairs is a function if it passes the vertical line test.

Because there are no more than one corresponding value for any given value, the relation of ordered pairs IS a function.

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

the range of f is :{7}.

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

. Since for all real numbers. To make the square root positive we need to have .

Therefore the domain is :

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.

We have We have for all real numbers. when . The denominator is undefined when .

The nominator is defined if .

The domain is:

is defined when . Since we do not want to be 0 in the denominator we must have . when x=2.

Thus we need to exclude 2 also. Therefore the domain is:

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: