Understanding the Concept of Zero in a Function

Oct 13, 2025 by miodeya.com 62 views

Understanding the Concept of Zero in a Function

In mathematics, a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). The zero of a function is the value of the input that results in an output of zero. In other words, it is the value of the input that makes the function equal to zero.

The Given Function

The given function is $f(x) = 3 \sqrt{x+3} - 6$ . To find the zero of this function, we need to set the function equal to zero and solve for x.

Setting the Function Equal to Zero

To find the zero of the function, we set the function equal to zero:

f(x) = 3 \sqrt{x+3} - 6 = 0

Solving for x

To solve for x, we can start by isolating the square root term:

3 \sqrt{x+3} = 6

Next, we can divide both sides by 3:

\sqrt{x+3} = 2

Now, we can square both sides to get rid of the square root:

(\sqrt{x+3})^2 = 2^2

This simplifies to:

x+3 = 4

Subtracting 3 from both sides gives us:

x = 1

Conclusion

Therefore, the zero of the function $f(x) = 3 \sqrt{x+3} - 6$ is $x = 1$ .

Answer

The correct answer is:

B. $x = 1$

Discussion

This problem requires the application of algebraic techniques to solve for the zero of a function. The key steps involved in solving this problem are:

Setting the function equal to zero
Isolating the square root term
Squaring both sides to get rid of the square root
Solving for x

By following these steps, we can find the zero of the function and determine the correct answer.

Additional Tips and Tricks

When solving for the zero of a function, it is essential to set the function equal to zero and solve for x.
Isolating the square root term can help simplify the equation and make it easier to solve.
Squaring both sides of the equation can help get rid of the square root, but it can also introduce extraneous solutions. Therefore, it is crucial to check the solutions carefully to ensure that they are valid.

Real-World Applications

The concept of finding the zero of a function has numerous real-world applications. For example:

In physics, the zero of a function can represent the equilibrium point of a system.
In engineering, the zero of a function can represent the point at which a system reaches a steady state.
In economics, the zero of a function can represent the point at which a system reaches a stable equilibrium.

Frequently Asked Questions

Q: What is the zero of a function?

A: The zero of a function is the value of the input that results in an output of zero. In other words, it is the value of the input that makes the function equal to zero.

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

A: To find the zero of a function, you need to set the function equal to zero and solve for x. This involves isolating the variable x and solving for its value.

Q: What are some common techniques for finding the zero of a function?

A: Some common techniques for finding the zero of a function include:

Setting the function equal to zero and solving for x
Isolating the square root term
Squaring both sides of the equation to get rid of the square root
Using algebraic techniques such as factoring and canceling

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

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

Not setting the function equal to zero
Not isolating the variable x
Squaring both sides of the equation without checking for extraneous solutions
Not checking the solutions carefully to ensure that they are valid

Q: How do I check if a solution is valid?

A: To check if a solution is valid, you need to plug the solution back into the original equation and check if it is true. If the solution is not true, then it is an extraneous solution and should be discarded.

Q: What are some real-world applications of finding the zero of a function?

A: Some real-world applications of finding the zero of a function include:

Finding the equilibrium point of a system in physics
Finding the steady state of a system in engineering
Finding the stable equilibrium of a system in economics

Q: Can you give an example of finding the zero of a function?

A: Yes, here is an example of finding the zero of a function:

Suppose we have the function f(x) = 3x^2 - 6x + 2. To find the zero of this function, we need to set the function equal to zero and solve for x:

3x^2 - 6x + 2 = 0

We can start by factoring the left-hand side of the equation:

(3x - 2)(x - 1) = 0

This tells us that either 3x - 2 = 0 or x - 1 = 0. Solving for x in each case, we get:

x = 2/3 or x = 1

Therefore, the zeros of the function are x = 2/3 and x = 1.

Q: Can you give another example of finding the zero of a function?

A: Yes, here is another example of finding the zero of a function:

Suppose we have the function f(x) = x^2 + 4x + 4. To find the zero of this function, we need to set the function equal to zero and solve for x:

x^2 + 4x + 4 = 0

We can start by factoring the left-hand side of the equation:

(x + 2)^2 = 0

This tells us that x + 2 = 0. Solving for x, we get:

x = -2

Therefore, the zero of the function is x = -2.

Q: What are some common types of functions that have zeros?

A: Some common types of functions that have zeros include:

Quadratic functions
Cubic functions
Polynomial functions
Rational functions

Q: Can you give an example of a function that does not have a zero?

A: Yes, here is an example of a function that does not have a zero:

Suppose we have the function f(x) = x^2 + 1. This function does not have a zero because there is no value of x that makes the function equal to zero.

Q: What are some common mistakes to avoid when working with functions that have zeros?

A: Some common mistakes to avoid when working with functions that have zeros include:

Not checking if the function has a zero
Not finding all the zeros of the function
Not checking if the solutions are valid

Q: How do I know if a function has a zero?

A: To determine if a function has a zero, you need to set the function equal to zero and solve for x. If you find a value of x that makes the function equal to zero, then the function has a zero at that value of x.