To solve this problem, we look for an operation to perform on both sides that will leave n^-2 by itself on one side. Dividing both sides by 25n^-3 would leave n^-2 by itself on the right side of the equqation, as shown below:

100n³p^–1/25n^–3 = 25n/25n^–3

Remember that when dividing terms with the same base, we subtract the exponents, so the equation can be written as 100n⁰p^–1/25 = n^–2

Finally, we simplify to find 4p^–1 = n^–2.  

Setting f(x)=10 and taking the equation out of the absolute value you get 10=3x-2 and -10=3x-2. Solving both of these equations for x gives you x=4 or -8/3.

For this question, we "plug in" the value of x given, which is inside the parentheses, and follow along the table to see what value the f or g functions output. For g(12), the output value is –1, while for f(9), the output value is 4 (be careful not to reverse these!) Thus, we can plug into the equation given:

(–1) – √4) = –1 – 2 = –3.

f(x, y) is defined as x²y² – xy + y. In order to find f(a, b), we will need to first find a and then b.

We are told that a = f(1, 3). We can use the definition of f(x, y) to determine the value of a.

a = f(1, 3) = 1²3² – 1(3) + 3 = 1(9) – 3 + 3 = 9 + 0 = 9

a = 9

Similarly, we can find b by determining the value of f(–2, –1).

b = f(–2, –1) = (–2)²(–1)² – (–2)(–1) + –1 = 4(1) – (2) – 1 = 4 – 2 – 1 = 1

b = 1

Now, we can find f(a, b), which is equal to f(9, 1).

f(a, b) = f(9, 1) = 9²(1²) – 9(1) + 1 = 81 – 9 + 1 = 73

f(a, b) = 73

The answer is 73.

Subtract the first expression from the second. That gives you 6z = 6. That simplifies to z = 1.

In this question, the first 5 terms of the sequence are given.  In other words, when the inputs n = {1, 2, 3, 4, 5}, an underlying function f(n) produces an output of {8, 11, 14, 17, 20}.

The best way to solve this problem is to use the answers.

The answers have 5 possible functions for this sequence.

2n + 4

2n + 5

3n + 4

3n + 5

4n + 4

When n = {1, 2, 3, 4, 5}, these functions produce the following sequence:

2n + 4 = {6, 8, 10, 12, 14}

2n + 5 = {7, 9, 11, 13, 15}

3n + 4 = {7, 10, 13, 16, 19}

3n + 5 = {8, 11, 14, 17, 20}

4n + 4 = {8, 12, 16, 20, 24}

Therefore the function 3n + 5 is the correct answer.

Note: when determining if a certain function is the correct one, you do not have to solve for all 5 terms.  As soon as one of the terms do not match the original sequence, that function can be eliminated.

Remember that a root of a function is the point where it crosses the x-axis, i.e. an x-intercept. The x-intercepts of a function occur where the y-value of a point is equal to zero. Therefore, the solutions to the equation f(x) = 0 give us the roots of f(x). Thus, if we set the functions in the answer choices equal to 0, and we end up solving the equation f(x) = 0, then the function will have the same roots as f(x) .

Let's set each of the functions in the answer choices equal 0.

First, let's look at (f(x))^(1/2) = 0. Raising a function to the 1/2 power is the same as taking the square root of f(x). To get rid of the square root, we can square both sides.

(f(x))^(1/2) = √(f(x)) = 0

(√f(x))² = 0²

f(x) = 0. Solving the equation f(x) = 0 will give us the roots of f(x). This means, to find the roots of (f(x))^(1/2), we will end up having to find the roots of f(x). In short, the roots of (f(x))^(1/2) will be the same as those of f(x).

Intuitively, this makes sense. When we graph (f(x))^(1/2) , what we are doing is taking the square root of the y-values of every point on f(x). The roots of f(x) will all have y-values of zero, and taking the square root of zero isn't going to change the y-value. Thus, the location of the roots on f(x) will not change, and the roots of (f(x))^(1/2) will be the same as the roots of f(x).

Next, we can look at (f(x))². We can set it equal to zero to find the roots.

(f(x))² = 0

We can then take the square root of both sides. The square root of zero is zero.

f(x) = 0

Once again, we are left with the equation f(x) = 0, which will give us the roots of f(x). This makes sense, because (f(x))² means we are squaring the y-value of the roots on f(x), which wouldn't change the location of the roots, because the square of zero is still zero.

The third graph is –4f(x). Once again, we can set it equal to zero.

–4f(x) = 0

Divide both sides by –4.

f(x) = 0

We are left with this equation, which will give us the roots of f(x). This makes sense, because if we were to multiply the y-values of the x-intercepts on f(x) by –4, then the location of the points wouldn't change. Multiply 0 by –4 will still give us 0.

The next function is |f(x)|. Whenever we take an absolute value of a function, we take any negative points and make them positive. Essentially, |f(x)| reflects all of the negative points on f(x) across the x-axis. However, points located on the x-axis are unchanged by taking the absolute value, because the absolute value of zero is still going to be zero. In other words, if we take the absolute value of the zeros on f(x), they will still be at the same location. Thus, |f(x)| and f(x) have the same roots.

This leaves the function (f(x))^–1. We can set it equal to zero to find its roots. Remember that, in general, a^–1 = 1/a.

(f(x))^–1 = 1/f(x) = 0

Multiply by f(x) on both sides.

1 = 0(f(x)) = 0

Clearly, we have something strange here, because one doesn't equal zero. This tells us that (f(x))^–1 can't have the same roots as f(x).

Let's look at an example of why the roots of f(x) and (f(x))^–1 won't necessarily be the same. Let f(x) = x. The only root of f(x) would be at the point (0,0). Next, let's look at (f(x))^–1 = 1/x. The graph of 1/x won't have any roots, because it has a vertical asymptote at zero (1/x is not defined when x = 0). Thus, f(x) doesn't have the same roots as (f(x))^–1. Intuitively, this makes since, because (f(x))^–1 essentially takes the reciprocal of the y-values on every point of f(x). But zero doesn't have a reciprocal (because 1/0 isn't defined), so the roots will likely change.

The answer is (f(x))^–1 .

You use the equation to solve for z, which equals 4. We then plug this in to f(3z), making x = 12, then add 4 as in the original equation, giving you 16.

We are told that f(x) must pass through the points (–1,5), (1,–1), and (2,–16). In order for it to pass through (–1,5), when x = –1 is input into the equation of f(x), the result must be 5. Likewise, when 1 and 2 are input into the equation of f(x), the results should be –1 and –16, respectively. To check if the equation for f(x) is correct, we need to put x = –1, 1, and 2 into the equation and see if we get the correct results. In short, we need to find the equation of f(x) such that f(–1) = 5, f(1) = –1, and f(2) = –16.

The only equation for f(x) that will pass through all three points is f(x) = 2x² – 3x³ . We can see this by testing x = –1, 1, and 2.

f(x) = 2x² – 3x³

f(–1) = 2(–1)² – 3(–1)³ = 2(1) – 3(–1) = 5

This means that (–1,5) would indeed lie on the graph of f(x).

f(1) = 2(1)² – 3(1)³ = 2 – 3 = –1

So f(x) would also pass through (1,–1).

f(2) = 2(2)² – 3(2)³ = 8 – 3(8) = –16

f(x) would also pass through (2,16).

We can check to see why the other equations of f(x) wouldn't satisfy all three points.

Let's look at the equation f(x) = –3x + 2 and test x = 2.

f(2) = –3(2) + 2 = –4, which doesn't equal –16. So that means this equation wouldn't pass through (2,–16).

Next, we can try f(x) = –1.5x² – 3x + 3.5, using x = 2.

f(2) = –1.5(2²) – 3(2) + 3.5 = –8.5, which is not –16.

Next, we can try f(x) = 1.5x² – 3x + 0.5 at x = 2.

f(2) = 1.5(2²) – 3(2) + 0.5 = 0.5, which is not equal to –16.

Finally, let's try f(x) = –x³ – 4x, using x = 1.

f(1) = –(1³) – 4(1) = –5, which doesn't equal –1.

The answer is f(x) = 2x² – 3x³ .

Let's go through each statement and determine if it's true.

f(x) = ax²

f(–x) = a(–x)² = a(–x)(–x) = ax² = f(x)

Statement I is true. Now let's look at the next statement.

f(x) = ax²

|f(x)| = |ax²|

Because a is positive, and because x² ≥ 0, the product of a and x² must be greater than or equal to zero. In other words, ax² ≥ 0. If we take the absolute value of a quantity that is greater than or equal to zero, we get that quantity. In general, if b ≥ 0, then |b| = b. This means that |ax²| = ax². Thus, we can write the following:

|f(x)| = |ax²| = ax² = f(x)

Therefore, statement II is also true. Let's look at the last statement.

f(f(x)) = f(ax²)= a(ax²)² = a(ax²)(ax²) = a³x⁴

This means III is also true.

The answer is I, II, and III.