Given The Functions $f(x)=x$ And $g(x)=1$, What Is The Domain Of $\left(\frac{g}{f}\right)(x$\]?A. $x \neq 0$B. $x \neq -1$C. All Real Numbers

Domain of a Function: Understanding the Concept

When dealing with functions, it's essential to understand the concept of the domain. The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it's the set of all possible x-values that the function can accept without resulting in an undefined or imaginary output.

Understanding the Given Functions

In this problem, we are given two functions:

• $f(x) = x$
• $g(x) = 1$

These functions are simple and straightforward. The function $f(x) = x$ is an identity function, which means it returns the input value unchanged. The function $g(x) = 1$ is a constant function, which always returns the value 1, regardless of the input.

The Composite Function

We are asked to find the domain of the composite function $\left(\frac{g}{f}\right)(x)$. To find the domain of this composite function, we need to understand what it means to divide two functions.

When we divide two functions, we are essentially finding the ratio of their outputs for a given input. In this case, we are dividing the function $g(x)$ by the function $f(x)$.

Finding the Domain

To find the domain of the composite function $\left(\frac{g}{f}\right)(x)$, we need to consider the values of x for which the function is defined.

Since $f(x) = x$, we know that $f(x)$ is defined for all real numbers. However, when we divide $g(x)$ by $f(x)$, we need to ensure that $f(x)$ is not equal to zero, as division by zero is undefined.

Therefore, the domain of the composite function $\left(\frac{g}{f}\right)(x)$ is all real numbers except for $x = 0$.

Conclusion

In conclusion, the domain of the composite function $\left(\frac{g}{f}\right)(x)$ is all real numbers except for $x = 0$. This is because $f(x)$ is defined for all real numbers, but we need to exclude $x = 0$ to avoid division by zero.

Answer

The correct answer is:

A. $x \neq 0$

Final Thoughts

Understanding the concept of the domain of a function is crucial in mathematics. It helps us to identify the values of x for which a function is defined and to avoid undefined or imaginary outputs. In this problem, we saw how to find the domain of a composite function by considering the values of x for which the function is defined.

Domain of a Function: Key Takeaways

• The domain of a function is the set of all possible input values (x-values) for which the function is defined.
• To find the domain of a composite function, we need to consider the values of x for which the function is defined.
• Division by zero is undefined, so we need to exclude values of x that would result in division by zero.

Common Mistakes to Avoid

• Failing to consider the values of x for which the function is defined.
• Not excluding values of x that would result in division by zero.

Real-World Applications

Understanding the concept of the domain of a function has many real-world applications. For example, in physics, the domain of a function can represent the range of possible values for a physical quantity, such as temperature or pressure. In engineering, the domain of a function can represent the range of possible values for a design parameter, such as the length of a beam or the diameter of a pipe.

Conclusion

In conclusion, the domain of a function is a crucial concept in mathematics that helps us to identify the values of x for which a function is defined. By understanding the concept of the domain, we can avoid undefined or imaginary outputs and ensure that our functions are well-defined and meaningful.
Domain of a Function: Q&A

In this article, we will continue to explore the concept of the domain of a function through a series of questions and answers.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined.

Q: Why is it important to understand the domain of a function?

A: Understanding the domain of a function is crucial because it helps us to identify the values of x for which the function is defined and to avoid undefined or imaginary outputs.

Q: How do I find the domain of a composite function?

A: To find the domain of a composite function, you need to consider the values of x for which the function is defined. This may involve excluding values of x that would result in division by zero or other undefined operations.

Q: What is the difference between the domain and the range of a function?

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range of a function is the set of all possible output values (y-values) that the function can produce.

Q: Can a function have a domain of all real numbers?

A: Yes, a function can have a domain of all real numbers. For example, the function f(x) = x has a domain of all real numbers.

Q: Can a function have a domain of only positive real numbers?

A: Yes, a function can have a domain of only positive real numbers. For example, the function f(x) = 1/x has a domain of only positive real numbers.

Q: How do I determine the domain of a function with a square root?

A: When a function has a square root, you need to ensure that the expression inside the square root is non-negative. This may involve excluding values of x that would result in a negative expression inside the square root.

Q: How do I determine the domain of a function with a fraction?

A: When a function has a fraction, you need to ensure that the denominator is not equal to zero. This may involve excluding values of x that would result in a denominator of zero.

Q: Can a function have a domain of only integers?

A: Yes, a function can have a domain of only integers. For example, the function f(x) = x^2 has a domain of only integers.

Q: Can a function have a domain of only rational numbers?

A: Yes, a function can have a domain of only rational numbers. For example, the function f(x) = x^2 has a domain of only rational numbers.

Q: How do I determine the domain of a function with a logarithm?

A: When a function has a logarithm, you need to ensure that the expression inside the logarithm is positive. This may involve excluding values of x that would result in a negative expression inside the logarithm.

Q: Can a function have a domain of all complex numbers?

A: Yes, a function can have a domain of all complex numbers. For example, the function f(z) = z has a domain of all complex numbers.

Conclusion

In conclusion, the domain of a function is a crucial concept in mathematics that helps us to identify the values of x for which a function is defined. By understanding the concept of the domain, we can avoid undefined or imaginary outputs and ensure that our functions are well-defined and meaningful.

Common Mistakes to Avoid

• Failing to consider the values of x for which the function is defined.
• Not excluding values of x that would result in division by zero or other undefined operations.
• Assuming that a function has a domain of all real numbers without checking.
• Failing to consider the domain of a composite function.

Real-World Applications

Understanding the concept of the domain of a function has many real-world applications. For example, in physics, the domain of a function can represent the range of possible values for a physical quantity, such as temperature or pressure. In engineering, the domain of a function can represent the range of possible values for a design parameter, such as the length of a beam or the diameter of a pipe.

Conclusion

In conclusion, the domain of a function is a crucial concept in mathematics that helps us to identify the values of x for which a function is defined. By understanding the concept of the domain, we can avoid undefined or imaginary outputs and ensure that our functions are well-defined and meaningful.