Understanding the Concept of Zeros in a Function

Oct 13, 2025 by windailys.com 72 views

Understanding the Concept of Zeros in a Function

In mathematics, the zeros of a function are the values of the variable (in this case, x) that make the function equal to zero. In other words, they are the solutions to the equation F(x) = 0. To find the zeros of a function, we need to set the function equal to zero and solve for x.

The Given Function

The given function is F(x) = \frac{(x+3)(x-1)}{(x-2)(x+2)}. To find the zeros of this function, we need to set it equal to zero and solve for x.

Setting the Function Equal to Zero

To set the function equal to zero, we need to multiply both sides of the equation by the denominator, which is (x-2)(x+2). This will eliminate the fraction and make it easier to solve for x.

F(x) = 0 \frac{(x+3)(x-1)}{(x-2)(x+2)} = 0

Multiplying Both Sides by the Denominator

Multiplying both sides of the equation by the denominator, we get:

(x+3)(x-1) = 0

Solving for x

To solve for x, we need to find the values of x that make the equation true. In this case, we have a product of two binomials equal to zero. This means that either the first binomial (x+3) is equal to zero, or the second binomial (x-1) is equal to zero.

Case 1: x+3 = 0

Solving for x, we get:

x + 3 = 0 x = -3

Case 2: x-1 = 0

Solving for x, we get:

x - 1 = 0 x = 1

Checking the Solutions

Now that we have found the possible solutions, we need to check if they are actually zeros of the function. To do this, we need to plug each solution back into the original function and check if it equals zero.

Plugging x = -3 into the Function

F(-3) = \frac{(-3+3)(-3-1)}{(-3-2)(-3+2)} F(-3) = \frac{(0)(-4)}{(-5)(-1)} F(-3) = 0

Plugging x = 1 into the Function

F(1) = \frac{(1+3)(1-1)}{(1-2)(1+2)} F(1) = \frac{(4)(0)}{(-1)(3)} F(1) = 0

Conclusion

Based on our analysis, we have found that the zeros of the function F(x) = \frac{(x+3)(x-1)}{(x-2)(x+2)} are x = -3 and x = 1.

Answer Key

The correct answers are:

A. -2: INCORRECT
B. 1: CORRECT
C. -5: INCORRECT
D. -3: CORRECT
E. 2: INCORRECT
F. 3: INCORRECT

Discussion

What are the zeros of the function F(x) = \frac{(x+3)(x-1)}{(x-2)(x+2)}? Check all that apply.

A. -2
B. 1
C. -5
D. -3
E. 2
F. 3

Step 1: Understand the concept of zeros in a function

The zeros of a function are the values of the variable (in this case, x) that make the function equal to zero.

Step 2: Set the function equal to zero

F(x) = 0 \frac{(x+3)(x-1)}{(x-2)(x+2)} = 0

Step 3: Multiply both sides by the denominator

(x+3)(x-1) = 0

Step 4: Solve for x

x + 3 = 0 or x - 1 = 0

Step 5: Check the solutions

F(-3) = 0 and F(1) = 0

Step 6: Conclusion

The zeros of the function F(x) = \frac{(x+3)(x-1)}{(x-2)(x+2)} are x = -3 and x = 1.

Step 7: Answer Key

The correct answers are:

A. -2: INCORRECT
B. 1: CORRECT
C. -5: INCORRECT
D. -3: CORRECT
E. 2: INCORRECT
F. 3: INCORRECT
Q&A: Zeros of a Function ==========================

Q: What are the zeros of a function?

A: The zeros of a function are the values of the variable (in this case, x) that make the function equal to zero.

Q: How do I find the zeros of a function?

A: To find the zeros of a function, you need to set the function equal to zero and solve for x. This can be done by multiplying both sides of the equation by the denominator, which will eliminate the fraction and make it easier to solve for x.

Q: What is the first step in finding the zeros of a function?

A: The first step in finding the zeros of a function is to set the function equal to zero. This is done by writing the equation F(x) = 0 and then multiplying both sides by the denominator.

Q: How do I know if a solution is a zero of the function?

A: To know if a solution is a zero of the function, you need to plug the solution back into the original function and check if it equals zero. If it does, then the solution is a zero of the function.

Q: What are some common mistakes to avoid when finding the zeros of a function?

A: Some common mistakes to avoid when finding the zeros of a function include:

Not setting the function equal to zero before solving for x
Not multiplying both sides of the equation by the denominator
Not checking the solutions to see if they are actually zeros of the function

Q: Can you give an example of how to find the zeros of a function?

A: Let's say we have the function F(x) = \frac{(x+3)(x-1)}{(x-2)(x+2)}. To find the zeros of this function, we would set it equal to zero and solve for x.

F(x) = 0 \frac{(x+3)(x-1)}{(x-2)(x+2)} = 0

Multiplying both sides by the denominator, we get:

(x+3)(x-1) = 0

Solving for x, we get:

x + 3 = 0 or x - 1 = 0

x = -3 or x = 1

Checking the solutions, we find that F(-3) = 0 and F(1) = 0. Therefore, the zeros of the function are x = -3 and x = 1.

Q: What is the significance of finding the zeros of a function?

A: Finding the zeros of a function is important because it helps us understand the behavior of the function. The zeros of a function are the points where the function crosses the x-axis, and they can give us information about the function's shape and behavior.

Q: Can you give some real-world examples of how finding the zeros of a function is used?

A: Yes, finding the zeros of a function is used in many real-world applications, such as:

Physics: Finding the zeros of a function can help us understand the behavior of physical systems, such as the motion of objects under the influence of gravity or friction.
Engineering: Finding the zeros of a function can help us design and optimize systems, such as electronic circuits or mechanical systems.
Economics: Finding the zeros of a function can help us understand the behavior of economic systems, such as the behavior of supply and demand curves.

Step 1: Understand the concept of zeros in a function

The zeros of a function are the values of the variable (in this case, x) that make the function equal to zero.

Step 2: Set the function equal to zero

F(x) = 0 \frac{(x+3)(x-1)}{(x-2)(x+2)} = 0

Step 3: Multiply both sides by the denominator

(x+3)(x-1) = 0

Step 4: Solve for x

x + 3 = 0 or x - 1 = 0

Step 5: Check the solutions

F(-3) = 0 and F(1) = 0

Step 6: Conclusion

The zeros of the function F(x) = \frac{(x+3)(x-1)}{(x-2)(x+2)} are x = -3 and x = 1.

Step 7: Answer Key

The correct answers are:

A. -2: INCORRECT
B. 1: CORRECT
C. -5: INCORRECT
D. -3: CORRECT
E. 2: INCORRECT
F. 3: INCORRECT