# Precalculus : Functions

## Example Questions

### Example Question #1 : Determine If A Relation Is A Function

Which of the following expressions is not a function?

Explanation:

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:

### Example Question #2 : Determine If A Relation Is A Function

Suppose we have the relation  on the set of real numbers  whenever . Which of the following is true.

The relation is a function because  holds and  also holds.

The relation is not a function because  and  both hold.

The relation is a function because every relation is a function, since that's how relations are defined.

The relation is not a function because  holds but  does not.

The relation is a function because for every , there is only one  such that  holds.

The relation is not a function because  and  both hold.

Explanation:

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.

### Example Question #3 : Determine If A Relation Is A Function

Consider a family consisting of a two parents, Juan and Oksana, and their daughters Adriana and Laksmi. A relation  is true whenever  is the child of . Which of the following is not true?

If the two parents had only one daughter, the relation would be a function.

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

The relation is not a function because (Laksmi,Juan) and (Laksmi,Oksana) both hold.

(Adriana,Laksmi) does not hold because Laksmi is not Adriana's child and is a boy.

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

Explanation:

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.

### Example Question #4 : Determine If A Relation Is A Function

Which of the following relations is not a function?

Explanation:

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.

### Example Question #5 : Determine If A Relation Is A Function

Given the set of ordered pairs, determine if the relation is a function

Cannot be determined

No

Yes

No

Explanation:

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.

### Example Question #6 : Determine If A Relation Is A Function

What equation is perpendicular to  and passes throgh ?

Explanation:

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 :

### Example Question #1 : Determine If A Relation Is A Function

Is the following relation of ordered pairs a function?

Yes

Cannot be determined

No

Yes

Explanation:

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.

### Example Question #71 : Functions

Which of the following is not true about a field. (Note: the real numbers  is a field)

We have  for any  and  in the field.

For every element  in the field, there is another element  such that their sum  is equal to , where  is the additive identity.

A field can be defined in many ways.

For every element  in the field, there is another element  such that their product   is equal to , where  is the multiplicative identity,  in the case of real numbers.

There is an element  in the field such that  for any element  in the field.

For every element  in the field, there is another element  such that their product   is equal to , where  is the multiplicative identity,  in the case of real numbers.

Explanation:

It is not the case that for any element  in a field, there is another one  such that their product is . Take  in the real numbers. Multiply  by any number and you get , so you will never get . This is true for any field that has more than 1 element.

### Example Question #1 : Modeling

John lives in Atlanta, but commutes every Monday to LaGrange where he has an apartment he stays in Monday-Friday for work. Each Monday he drives 350 miles to LaGrange. Once he arrives to his home away from home he is in walking distance of work and does not use his car for anything else. After 23 weeks his odometer shows 186,000 miles. Write an equation that models his odometer reading as a function of the number of weeks he has been driving after commencing his new job.

Explanation:

The rate of change of his mileage is 700 per week (350 x 2=700 there and back). The rate of change is the same thing as slope. Since we are looking for equation an equation that models his odometer reading as a function of the number of weeks he has been driving we can extract the point (23 , 18600) since after 23 weeks his odometer read 18,600 miles. Now we will use the point slope formula:

distribute the right side

isolate y

### Example Question #2 : Modeling

John lives in Atlanta, but commutes every Monday to LaGrange where he has an apartment he stays in Monday-Friday. Each Monday he drives 350 miles to LaGrange. Once he arrives to his apartment he is in walking distance of work and does not use his car for anything else. After 23 weeks his odometer shows 186,000 miles. Write an equation that models his odometer reading as a function of the number of weeks he has been driving after commencing his new job. Using the equation you just made, what is the y intercept or his original mileage before starting?

y intercept=16,100 miles

y intercept=186,000 miles

y intercept=169,900 miles