### All SAT II Math I Resources

## Example Questions

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

Define .

Give the range of .

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

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

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

### Example Question #1 : Functions And Graphs

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

Give the natural domain of the composite function .

**Possible Answers:**

The set of all real numbers

**Correct answer:**

The set of all real numbers

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 #1 : Functions And Graphs

**Possible Answers:**

Domain: All real numbers

Range:

Domain: All real numbers

Range:

**Correct answer:**

Domain: All real numbers

Range:

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 : Domain And Range

Which of the following is NOT a function?

**Possible Answers:**

**Correct answer:**

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 #1 : How To Find The Domain Of A Function

Give the domain of the function below.

**Possible Answers:**

**Correct answer:**

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 : How To Factor The Quadratic Equation

Find the domain:

**Possible Answers:**

**Correct answer:**

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 #1 : Functions And Graphs

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

**Possible Answers:**

**Correct answer:**

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 #1 : Domain And Range

What is the range of the function?

**Possible Answers:**

**Correct answer:**

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 #1 : Domain And Range

What is the domain of the function?

**Possible Answers:**

**Correct answer:**

The domain represents the acceptable values for this function. Based on the members of the function, the only limit that you have is the non-allowance of a negative number (because of the square root). The square and the linear terms are fine with any numbers. You cannot have any negative values, otherwise the square root will not be a real number.

Minimum: 0 inclusive, maximum: infinity

Domain:

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