# SAT II Math I : Properties of Functions and Graphs

## Example Questions

### Example Question #1 : Range And Domain

Define .

Give the range of

Explanation:

The radicand within a square root symbol must be nonnegative, so

This happens if and only if , so the domain of  is .

assumes its greatest value when , which is the point on  where  is least - this is at .

Similarly,  assumes its least value when , which is the point on  where  is greatest - this is at .

Therefore, the range of  is .

### Example Question #1 : Range And Domain

Define .

Give the domain of

Explanation:

The radicand within a square root symbol must be nonnegative, so

This happens if and only if , so the domain of  is .

### Example Question #2 : Range And Domain

Define the functions  and  on the set of real numbers as follows:

Give the natural domain of the composite function .

The set of all real numbers

The set of all real numbers

Explanation:

The natural domain of the composite function  is defined to be the intersection two sets.

One set is the natural domain of . Since  is a polynomial, its domain is the set of all real numbers.

The other set is the set of all values of  such that that  is in the domain of . Since the radicand of the square root in  must be nonnegative,

, and , the domain of

Therefore, the other set is the set of all  such that

Substitute:

This holds for all real numbers, so this set is also the set of all real numbers.

The natural domain of  is the set of all real numbers.

### Example Question #5 : Functions And Graphs

is a sine curve. What are the domain and range of this function?

Domain: All real numbers

Range:

Domain: All real numbers

Range:

Domain: All real numbers

Range:

Explanation:

The domain includes the values that go into a function (the x-values) and the range are the values that come out (the  or y-values). A sine curve represent a wave the repeats at a regular frequency. Based upon this graph, the maximum  is equal to 1, while the minimum is equal to –1. The x-values span all real numbers, as there is no limit to the input fo a sine function. The domain of the function is all real numbers and the range is .

### Example Question #1 : Range And Domain

Which of the following is NOT a function?

Explanation:

A function has to pass the vertical line test, which means that a vertical line can only cross the function one time.  To put it another way, for any given value of , there can only be one value of .  For the function , there is one value for two possible  values.  For instance, if , then .  But if , as well.  This function fails the vertical line test.  The other functions listed are a line,, the top half of a right facing parabola, , a cubic equation, , and a semicircle, . These will all pass the vertical line test.

### Example Question #2 : Range And Domain

Give the domain of the function below.

Explanation:

The domain is the set of possible value for the variable. We can find the impossible values of by setting the denominator of the fractional function equal to zero, as this would yield an impossible equation.

Now we can solve for .

There is no real value of that will fit this equation; any real value squared will be a positive number.

The radicand is always positive, and is defined for all real values of . This makes the domain of  the set of all real numbers.

### Example Question #1 : Properties Of Functions And Graphs

Find the domain:

Explanation:

To find the domain, find all areas of the number line where the fraction is defined.

because the denominator of a fraction must be nonzero.

Factor by finding two numbers that sum to -2 and multiply to 1.  These numbers are -1 and -1.

### Example Question #6 : Functions And Graphs

If , which of these values of  is NOT in the domain of this equation?

Explanation:

Using  as the input () value for this equation generates an output () value that contradicts the stated condition of .

Therefore  is not a valid value for  and not in the equation's domain:

### Example Question #4 : Range And Domain

What is the range of the function?

Explanation:

This function is a parabola that has been shifted up five units. The standard parabola has a range that goes from 0 (inclusive) to positive infinity. If the vertex has been moved up by 5, this means that its minimum has been shifted up by five. The first term is inclusive, which means you need a "[" for the beginning.

Minimum: 5 inclusive, maximum: infinity

Range:

### Example Question #36 : Introduction To Functions

What is the domain of the function?