### All PSAT Math Resources

## Example Questions

### Example Question #1 : Algebraic Fractions

Given the expression, , which value CANNOT be equal to ?

**Possible Answers:**

**Correct answer:**

cannot equal a value which would make the denominator equal to 0. In order to figure out what that value is, we must first simplify this fraction, then set each factor of the denominator equal to 0. As follows:

First, simplify by finding the original binomial multiples:

Now, set each factor of the denominator equal to 0

If OR , the denominator will equal 0. is the only choice provided by the answers.

Note: Even though you can cancel out from the numerator and denominator, still cannot be equal to two. The graph would have a hole at and an aymptote at .

### Example Question #1 : Algebraic Fractions

Define the function as follows:

Give the domain of .

**Possible Answers:**

**Correct answer:**

The numerator, being a polynomial, is not restricting our domain. The domain is, however, restricted by the polynomial in the denominator, which must be nonzero. Therefore, we set the denominator equal to zero to determine the excluded values:

Therefore, the domain of is the set of all real numbers except - that is,

### Example Question #2 : Algebraic Fractions

Define the function as follows:

Give the domain of .

**Possible Answers:**

**Correct answer:**

The domain of is restricted by two different denominators, neither of which can be equal to 0, so the excluded values are:

The correct response is therefore .

### Example Question #3 : Algebraic Fractions

Define the function as follows:

Give the domain of .

**Possible Answers:**

**Correct answer:**

The numerator, being a polynomial, does not restrict our domain. The denominator, however, does restrict it to the values for which it is not equal to 0. We set the denominator equal to 0 to find the excluded values:

The domain, in interval notation, is therefore

.

### Example Question #5 : Algebraic Fractions

Define the function on the set of real numbers as follows:

Give the domain of .

**Possible Answers:**

**Correct answer:**

The domain of is restricted by two things.

First, the expression within the radical in the numerator must be nonnegative. We therefore solve for in the inequality

,

or, in interval notation,

Second, the expression in the denominator must be nonzero. Therefore, we set the denominator equal to zero to determine the excluded value(s):

We exclude 4 from , so the correct response is

### Example Question #4 : Algebraic Fractions

Define the function on the set of real numbers as follows:

Give the domain of .

**Possible Answers:**

**Correct answer:**

There are two things restricting the domain of .

One is the radical symbol in the numerator. The expression inside the radical must be nonnegative, so solve the inequality:

,

or, in interval notation,

The other is the denominator, which must be equal to 0, so set, and solve for in, the equation:

is already excluded from the domain; we exclude 4, so the domain is

.

### Example Question #7 : Algebraic Fractions

Define the function on the real numbers as follows:

Give the domain of .

**Possible Answers:**

**Correct answer:**

The numerator, being a polynomial, is not restricting our domain. The domain is, however, restricted by the expression in the denominator, which must be nonzero. Furthermore, the radicand must be nonnegative. Combined, these facts mean that the radicand must be positive, and that the following inequality be solved:

or, equivalently, .

In interval form, this is

### Example Question #8 : Algebraic Fractions

Define the functions and as follows:

and

Give the domain of the function .

**Possible Answers:**

**Correct answer:**

The domain of the product of two functions is the intersection of the domains of the individual functions.

The domain of is restricted to all values of that yield a nonzero denominator. Since this means that

,

then, subsequently,

,

so the domain is the set of all real numbers except 4.

Similarly, the domain of is restricted to all values of that yield a nonzero denominator. This set is found to be the set of all real numbers except 7.

The intersection of these sets is the set of all real numbers except 4 and 7, or

.

### Example Question #9 : Algebraic Fractions

Define the function as follows:

Give the domain of .

**Possible Answers:**

**Correct answer:**

The definition of has two denominators of two fractions (one within the other), so we must exclude the values of that make either denominator equal to zero.

One denominator is . Since it cannot be zero, we have

,

and, subsequently,

.

The value 2 is excluded from the domain.

The other denominator is . If it is equal to zero, then

Therefore, this value is also excluded from the domain.

The correct domain is the set .

### Example Question #5 : Algebraic Fractions

Define the function as follows:

Give the domain of .

**Possible Answers:**

**Correct answer:**

The numerator , being a polynomial, does not restrict the domain.

appears as a radicand of a square root in the definition of , so must be restricted to nonnegative numbers. Also, is a denominator, which means 0 must also be excluded. Therefore, we are restricted so far to positive numbers.

There is one more denominator, which is ; it must be nonzero. We set this equal to zero to determine any additional value(s) that must be excluded:

Therefore, the domain is the set of all positive numbers except 9 - or

.

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