Which Function Is Undefined For $x=0$?A. Y = X − 2 3 Y=\sqrt[3]{x-2} Y = 3 X − 2 ​ B. Y = X − 2 Y=\sqrt{x-2} Y = X − 2 ​ C. Y = X + 2 3 Y=\sqrt[3]{x+2} Y = 3 X + 2 ​ D. Y = X + 2 Y=\sqrt{x+2} Y = X + 2 ​

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Introduction

In mathematics, functions are used to describe the relationship between variables. However, not all functions are defined for all values of the variable. In this article, we will explore which function is undefined for x=0x=0 among the given options.

What is a Function?

A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. It is a rule that assigns to each input exactly one output. In other words, a function takes an input and produces an output.

Types of Functions

There are several types of functions, including:

  • Polynomial functions: These are functions that can be written in the form f(x)=anxn+an−1xn−1+…+a1x+a0f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0, where an≠0a_n \neq 0.
  • Rational functions: These are functions that can be written in the form f(x)=p(x)q(x)f(x) = \frac{p(x)}{q(x)}, where p(x)p(x) and q(x)q(x) are polynomials.
  • Trigonometric functions: These are functions that involve the trigonometric functions sine, cosine, and tangent.
  • Exponential functions: These are functions that involve the exponential function exe^x.
  • Root functions: These are functions that involve the nth root of a number.

Root Functions

Root functions are a type of function that involves the nth root of a number. They are defined as:

f(x)=xnf(x) = \sqrt[n]{x}

where nn is a positive integer.

Undefined Functions

A function is undefined when it is not possible to assign a value to the output for a given input. In other words, a function is undefined when it is not possible to evaluate the function at a particular point.

Which Function is Undefined for x=0x=0?

Now, let's examine the given options and determine which function is undefined for x=0x=0.

Option A: y=x−23y=\sqrt[3]{x-2}

This function is a root function with a positive integer exponent. It is defined for all real numbers, including x=0x=0.

Option B: y=x−2y=\sqrt{x-2}

This function is also a root function, but with a non-integer exponent. It is defined for all real numbers, including x=0x=0, but only if x−2≥0x-2 \geq 0, which means x≥2x \geq 2.

Option C: y=x+23y=\sqrt[3]{x+2}

This function is a root function with a positive integer exponent. It is defined for all real numbers, including x=0x=0.

Option D: y=x+2y=\sqrt{x+2}

This function is a root function with a non-integer exponent. It is defined for all real numbers, including x=0x=0, but only if x+2≥0x+2 \geq 0, which means x≥−2x \geq -2.

Conclusion

Based on the analysis above, we can conclude that the function y=x−2y=\sqrt{x-2} is undefined for x=0x=0 because it is not defined for x<2x < 2.

Final Answer

The final answer is B. y=x−2y=\sqrt{x-2}.

References

Additional Resources

Introduction

In our previous article, we explored which function is undefined for x=0x=0 among the given options. In this article, we will answer some frequently asked questions about undefined functions in mathematics.

Q: What is an undefined function?

A: An undefined function is a function that is not defined for a particular value of the variable. In other words, it is a function that does not have a value for a given input.

Q: Why are some functions undefined?

A: Some functions are undefined because they involve operations that are not defined for certain values of the variable. For example, the square root of a negative number is undefined in the real number system.

Q: Can a function be undefined for a single value of the variable?

A: Yes, a function can be undefined for a single value of the variable. For example, the function f(x)=1xf(x) = \frac{1}{x} is undefined for x=0x=0.

Q: Can a function be undefined for a range of values of the variable?

A: Yes, a function can be undefined for a range of values of the variable. For example, the function f(x)=xf(x) = \sqrt{x} is undefined for x<0x < 0.

Q: How do I determine if a function is undefined?

A: To determine if a function is undefined, you need to examine the function and identify any values of the variable that would cause the function to be undefined. You can do this by:

  • Checking the domain of the function
  • Looking for any restrictions on the variable
  • Checking for any values of the variable that would cause the function to be undefined

Q: What are some common examples of undefined functions?

A: Some common examples of undefined functions include:

  • The square root of a negative number
  • The reciprocal of zero
  • The logarithm of a non-positive number
  • The tangent of an odd multiple of π2\frac{\pi}{2}

Q: Can I define a function to be undefined for a particular value of the variable?

A: Yes, you can define a function to be undefined for a particular value of the variable. For example, you can define the function f(x)=1xf(x) = \frac{1}{x} to be undefined for x=0x=0.

Q: Why is it important to understand undefined functions?

A: Understanding undefined functions is important because it helps you to:

  • Avoid errors in your calculations
  • Understand the behavior of functions
  • Make informed decisions about the values of the variable

Conclusion

In this article, we have answered some frequently asked questions about undefined functions in mathematics. We have discussed what an undefined function is, why some functions are undefined, and how to determine if a function is undefined. We have also provided some common examples of undefined functions and discussed why understanding undefined functions is important.

Final Answer

The final answer is that understanding undefined functions is crucial in mathematics, and it helps you to avoid errors, understand the behavior of functions, and make informed decisions about the values of the variable.

References

Additional Resources