# SAT II Math II : Range and Domain

## Example Questions

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### Example Question #1 : Functions And Graphs

Define .

Give the range of .

The correct range is not among the other responses.

The correct range is not among the other responses.

Explanation:

The function can be rewritten as follows:

The expression  can assume any value except for 0, so the expression  can assume any value except for 1. The range is therefore the set of all real numbers except for 1, or

.

This choice is not among the responses.

### Example Question #1 : Functions And Graphs

Define .

Give the domain of .

Explanation:

In a rational function, the domain excludes exactly the value(s) of the variable which make the denominator equal to 0. Set the denominator to find these values:

The domain is the set of all real numbers except 7 - that is, .

### Example Question #1 : Functions And Graphs

Define

Give the domain of

Explanation:

Every real number has one real cube root, so there are no restrictions on the radicand of a cube root expression. The domain is the set of all real numbers.

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

Define

Give the range of .

Explanation:

for any real value of .

Therefore,

The range is .

### Example Question #1 : Range And Domain

Define

Give the range of .

Explanation:

for any real value of , so

,

making the range .

### Example Question #6 : Functions And Graphs

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 : Functions And Graphs

Define

Give the range of .

Explanation:

can be rewritten as .

For all real values of ,

or .

Therefore,

or  and

or  .

The range of  is .

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

What is the domain of the function

Explanation:

The domain of a function is all the x-values that in that function. The function  is a upward facing parabola with a vertex as (0,3). The parabola keeps getting wider and is not bounded by any x-values so it will continue forever. Parenthesis are used because infinity is not a definable number and so it can not be included.

### Example Question #2 : Functions And Graphs

What is the domain of the function?

Explanation:

Notice this function resembles the parent function .  The value of  must be zero or greater.

Set up an inequality to determine the domain of .

Subtract three from both sides.

Divide by negative ten on both sides.  The sign will switch.

The domain is:

### Example Question #10 : Functions And Graphs

What is the range of the function ?

All real numbers except .

All real numbers except .

All real numbers except .

All real numbers.