### All SAT Math Resources

## Example Questions

### Example Question #1 : How To Find Absolute Value

For which of the following functions below does *f*(*x*) = |*f*(*x*)| for every value of *x* within its domain?

**Possible Answers:**

f(x) = x^{4} + x

f(x) = x^{2} – 9

f(x) = x^{4} + (1 – x)^{2}

f(x) = x^{2} – 2x

f(x) = 2x + 3

**Correct answer:**

f(x) = x^{4} + (1 – x)^{2}

When we take the absolute value of a function, any negative values get changed into positive values. Essentially, |*f*(*x*)| will take all of the negative values of *f*(*x*) and reflect them across the *x*-axis. However, any values of *f*(*x*) that are positive or equal to zero will not be changed, because the absolute value of a positive number (or zero) is still the same number.

If we can show that *f*(*x*) has negative values, then |*f*(*x*)| will be different from *f*(*x*) at some points, because its negative values will be changed to positive values. In other words, our answer will consist of the function that never has negative values.

Let's look at *f*(*x*) = 2*x* + 3. Obviously, this equation of a line will have negative values. For example, where *x* = –4, *f*(–4) = 2(–4) + 3 = –5, which is negative. Thus, *f*(*x*) has negative values, and if we were to graph |*f*(*x*)|, the result would be different from *f*(*x*). Therefore, *f*(*x*) = 2*x* + 3 isn't the correct answer.

Next, let's look at *f*(*x*) = *x*^{2} – 9. If we let *x* = 1, then *f*(1) = 1 – 9 = –8, which is negative. Thus |*f*(*x*)| will not be the same as *f*(*x*), and we can eliminate this choice as well.

Now, let's examine *f*(*x*) = *x*^{2} – 2*x*. We know that *x*^{2} by itself can never be negative. However, if *x*^{2} is really small, then adding –2*x* could make it negative. Therefore, let's evaluate *f*(*x*) when *x* is a fractional value such as 1/2. *f*(1/2) = 1/4 – 1 = –3/4, which is negative. Thus, there are some values on *f*(*x*) that are negative, so we can eliminate this function.

Next, let's examine *f*(*x*) = *x*^{4} + *x*. In general, any number taken to an even-numbered power must always be non-negative. Therefore, *x*^{4} cannot be negative, because if we multiplied a negative number by itself four times, the result would be positive. However, the *x* term could be negative. If we let *x* be a small negative fraction, then *x*^{4} would be close to zero, and we would be left with *x*, which is negative. For example, let's find *f*(*x*) when *x* = –1/2. *f*(–1/2) = (–1/2)^{4 }+ (–1/2) = (1/16) – (1/2) = –7/16, which is negative. Thus, |*f*(*x*)| is not always the same as *f*(*x*).

By process of elimination, the answer is *f*(*x*) = *x*^{4} + (1 – *x*)^{2} . This makes sense because *x*^{4} can't be negative, and because (1 – *x*)^{2 }can't be negative. No matter what we subtract from one, when we square the final result, we can't get a negative number. And if we add *x*^{4} and (1 – *x*)^{2}, the result will also be non-negative, because adding two non-negative numbers always produces a non-negative result. Therefore, *f*(*x*) = *x*^{4} + (1 – *x*)^{2 }will not have any negative values, and |*f*(*x*)| will be the same as *f*(*x*) for all values of *x*. ^{}

The answer is *f*(*x*) = *x*^{4} + (1 – *x*)^{2} .

### Example Question #2 : How To Find Absolute Value

Let and both be negative numbers such that and . What is ?

**Possible Answers:**

**Correct answer:**

We need to solve the two equations |2a – 3| = 5 and |3 – 4b| = 11, in order to determine the possible values of a and b. When solving equations involving absolute values, we must remember to consider both the positive and negative cases. For example, if |x| = 4, then x can be either 4 or –4.

Let's look at |2a – 3| = 5. The two equations we need to solve are 2a – 3 = 5 and 2a – 3 = –5.

2a – 3 = 5 or 2a – 3 = –5

Add 3 to both sides.

2a = 8 or 2a = –2

Divide by 2.

a = 4 or a = –1

Therefore, the two possible values for a are 4 and –1. However, the problem states that both a and b are negative. Thus, a must equal –1.

Let's now find the values of b.

3 – 4b = 11 or 3 – 4b = –11

Subtract 3 from both sides.

–4b = 8 or –4b = –14

Divide by –4.

b = –2 or b = 7/2

Since b must also be negative, b must equal –2.

We have determined that a is –1 and b is –2. The original question asks us to find |b – a|.

|b – a| = |–2 – (–1)| = | –2 + 1 | = |–1| = 1.

The answer is 1.

### Example Question #1 : How To Find Absolute Value

**Possible Answers:**

**Correct answer:**

### Example Question #4 : How To Find Absolute Value

Find the absolute value of the following expression:

**Possible Answers:**

**Correct answer:**

In order the find the answer, you must first solve what is inside the absolute value signs.

Following order of operations, you must first multiply which equals .

Then you must subtract from as shown below:

Now, you must take the absolute value of which is positive , the correct answer.

### Example Question #2 : Absolute Value

Evaluate for :

**Possible Answers:**

**Correct answer:**

### Example Question #1 : Absolute Value

Evaluate for :

**Possible Answers:**

**Correct answer:**

Substitute 0.6 for :

### Example Question #8 : How To Find Absolute Value

Evaluate for :

**Possible Answers:**

**Correct answer:**

Substitute .

### Example Question #3 : How To Find Absolute Value

Which of the following sentences is represented by the equation

**Possible Answers:**

The sum of three and the absolute value of the sum of a number is three less than the number.

None of the other responses are correct.

The sum of three and the absolute value of the sum of a number is three greater than the number.

The absolute value of the sum of a number and seven is three greater than the number.

The absolute value of the sum of a number and seven is three less than the number.

**Correct answer:**

The absolute value of the sum of a number and seven is three less than the number.

is the absolute value of , which in turn is the sum of a number and seven and a number. Therefore, can be written as "the absolute value of the sum of a number and seven". Since it is equal to , it is three less than the number, so the equation that corresponds to the sentence is

"The absolute value of the sum of a number and seven is three less than the number."

### Example Question #1 : Absolute Value

Define

Evaluate .

**Possible Answers:**

None of the other responses is correct.

**Correct answer:**

### Example Question #691 : Arithmetic

Define an operation as follows:

For all real numbers ,

Evaluate: .

**Possible Answers:**

None of the other responses is correct.

The expression is undefined.

**Correct answer:**

, or, equivalently,