Functions - Pre-Calculus

To find the inverse of a function, replace the x any y positions:

Original Equation:

Inversed Equation:

Now solve for the inversed y value.

To find the inverse function of

we replace the with and vice versa.

So

Now solve for

Cancel the absolute value sign by separating the function into its positive and negative counterparts.

Evaluate the first scenario.

Evaluate the second scenario.

The correct answer is:

One way to approach this problem would be to plug in each answer and see what works. However, I would be a little more strategic and eliminate any options that don't make sense.

Our y value will never be negative, so eliminate any options with a negative y-value.

Try (0,0) really quick, since it's really easy

The only point that makes sense is (5,83), therefore it is the correct answer

We evaluate for

Since the absolute value of any number represents its magnitude from and is therefore always positive, the final answer would be

The greatest integer function takes an input and produces the greatest integer less than the input. Thus, the output is always smaller than the input and is an integer itself. Since our input was , we are looking for an integer less than this, which must be since any smaller integer would by definition not be "greatest".

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.