Describe The Domain For The Function $f(x)=\frac{\sqrt{x-6}}{x-7}$. Write Your Answer In Interval Notation.

Introduction

In mathematics, the domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of x that can be plugged into the function without causing any mathematical inconsistencies or undefined results. In this article, we will analyze the domain of the function $f(x)=\frac{\sqrt{x-6}}{x-7}$ and express it in interval notation.

Understanding the Function

The given function is a rational function, which means it is the ratio of two polynomials. The numerator of the function is $\sqrt{x-6}$, and the denominator is $x-7$. To determine the domain of the function, we need to consider the restrictions imposed by both the numerator and the denominator.

Restrictions Imposed by the Numerator

The numerator of the function is $\sqrt{x-6}$. Since the square root of a negative number is undefined in real numbers, we must have $x-6 \geq 0$. This implies that $x \geq 6$. Therefore, the domain of the function is restricted by the numerator to be $x \geq 6$.

Restrictions Imposed by the Denominator

The denominator of the function is $x-7$. Since division by zero is undefined, we must have $x-7 \neq 0$. This implies that $x \neq 7$. Therefore, the domain of the function is restricted by the denominator to be $x \neq 7$.

Combining the Restrictions

To determine the final domain of the function, we need to combine the restrictions imposed by both the numerator and the denominator. Since the numerator restricts the domain to be $x \geq 6$ and the denominator restricts the domain to be $x \neq 7$, the final domain of the function is the intersection of these two sets.

Interval Notation

The final domain of the function can be expressed in interval notation as $(6, 7) \cup (7, \infty)$. This notation indicates that the domain of the function consists of two separate intervals: $(6, 7)$ and $(7, \infty)$.

Conclusion

In conclusion, the domain of the function $f(x)=\frac{\sqrt{x-6}}{x-7}$ is $(6, 7) \cup (7, \infty)$. This means that the function is defined for all real numbers greater than or equal to 6, except for the value 7, which makes the denominator zero.

Key Takeaways

• The domain of a function is the set of all possible input values for which the function is defined.
• The domain of a function can be restricted by both the numerator and the denominator.
• The final domain of a function is the intersection of the restrictions imposed by both the numerator and the denominator.
• The domain of a function can be expressed in interval notation.

Final Answer

Introduction

In our previous article, we analyzed the domain of the function $f(x)=\frac{\sqrt{x-6}}{x-7}$ and expressed it in interval notation. In this article, we will answer some frequently asked questions related to the domain of this function.

Q: What is the domain of the function $f(x)=\frac{\sqrt{x-6}}{x-7}$?

A: The domain of the function $f(x)=\frac{\sqrt{x-6}}{x-7}$ is $(6, 7) \cup (7, \infty)$.

Q: Why is the domain restricted to $x \geq 6$?

A: The domain is restricted to $x \geq 6$ because the square root of a negative number is undefined in real numbers. Since the numerator of the function is $\sqrt{x-6}$, we must have $x-6 \geq 0$, which implies that $x \geq 6$.

Q: Why is the domain restricted to $x \neq 7$?

A: The domain is restricted to $x \neq 7$ because division by zero is undefined. Since the denominator of the function is $x-7$, we must have $x-7 \neq 0$, which implies that $x \neq 7$.

Q: How do we combine the restrictions imposed by both the numerator and the denominator?

A: To combine the restrictions imposed by both the numerator and the denominator, we need to find the intersection of the two sets. In this case, the numerator restricts the domain to be $x \geq 6$, and the denominator restricts the domain to be $x \neq 7$. Therefore, the final domain of the function is the intersection of these two sets, which is $(6, 7) \cup (7, \infty)$.

Q: What is the significance of the interval notation $(6, 7) \cup (7, \infty)$?

A: The interval notation $(6, 7) \cup (7, \infty)$ indicates that the domain of the function consists of two separate intervals: $(6, 7)$ and $(7, \infty)$. This means that the function is defined for all real numbers greater than or equal to 6, except for the value 7, which makes the denominator zero.

Q: Can we express the domain of the function in other notations?

A: Yes, we can express the domain of the function in other notations. For example, we can express the domain as $[6, \infty) \setminus \{7\}$ or $[6, \infty) - \{7\}$. However, the interval notation $(6, 7) \cup (7, \infty)$ is the most common and convenient way to express the domain of this function.

Q: How does the domain of the function affect its graph?

A: The domain of the function affects its graph by restricting the values of x for which the function is defined. In this case, the domain of the function is $(6, 7) \cup (7, \infty)$, which means that the graph of the function will be undefined at x = 7.

Conclusion

In conclusion, the domain of the function $f(x)=\frac{\sqrt{x-6}}{x-7}$ is $(6, 7) \cup (7, \infty)$. We hope that this Q&A article has helped to clarify any questions or doubts you may have had about the domain of this function.

Key Takeaways

• The domain of a function is the set of all possible input values for which the function is defined.
• The domain of a function can be restricted by both the numerator and the denominator.
• The final domain of a function is the intersection of the restrictions imposed by both the numerator and the denominator.
• The domain of a function can be expressed in interval notation.
• The domain of a function affects its graph by restricting the values of x for which the function is defined.

Final Answer

The final answer is $\boxed{(6, 7) \cup (7, \infty)}$.