Relations and Functions - Pre-Calculus

Recall that an expression is only a function if it passes the vertical line test. Test this by graphing each function and looking for one which fails the vertical line test. (The vertical line test consists of drawing a vertical line through the graph of an expression. If the vertical line crosses the graph of the expression more than once, the expression is not a function.)

Functions can only have one y value for every x value. The only choice that reflects this is:

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}.

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.