The Function $f$ Is Defined Below:$f(x) = \frac{x^2 - 12x + 32}{x^2 - 7x - 8}$Find All Values Of $x$ That Are NOT In The Domain Of $f$. If There Is More Than One Value, Separate Them With Commas.$\square$

Introduction

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs. The domain of a function is the set of all possible input values for which the function is defined. In this article, we will explore the function $f$ defined as $f(x) = \frac{x^2 - 12x + 32}{x^2 - 7x - 8}$ and find all values of $x$ that are not in the domain of $f$.

The Function $f$

The function $f$ is defined as $f(x) = \frac{x^2 - 12x + 32}{x^2 - 7x - 8}$. To find the values of $x$ that are not in the domain of $f$, we need to find the values of $x$ that make the denominator of the function equal to zero.

Finding the Values of $x$ That Make the Denominator Equal to Zero

To find the values of $x$ that make the denominator equal to zero, we need to solve the equation $x^2 - 7x - 8 = 0$. This is a quadratic equation, and we can solve it using the quadratic formula or by factoring.

Factoring the Quadratic Equation

We can factor the quadratic equation $x^2 - 7x - 8 = 0$ as $(x - 8)(x + 1) = 0$. This gives us two possible values for $x$: $x = 8$ and $x = -1$.

Solving the Quadratic Equation Using the Quadratic Formula

We can also solve the quadratic equation $x^2 - 7x - 8 = 0$ using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In this case, $a = 1$, $b = -7$, and $c = -8$. Plugging these values into the formula, we get:

$x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(-8)}}{2(1)}$ $x = \frac{7 \pm \sqrt{49 + 32}}{2}$ $x = \frac{7 \pm \sqrt{81}}{2}$ $x = \frac{7 \pm 9}{2}$

This gives us two possible values for $x$: $x = \frac{7 + 9}{2} = 8$ and $x = \frac{7 - 9}{2} = -1$.

Conclusion

In conclusion, the values of $x$ that are not in the domain of $f$ are $x = 8$ and $x = -1$. These values make the denominator of the function equal to zero, and therefore, they are not in the domain of $f$.

Final Answer

Introduction

In our previous article, we explored the function $f$ defined as $f(x) = \frac{x^2 - 12x + 32}{x^2 - 7x - 8}$ and found the values of $x$ that are not in the domain of $f$. In this article, we will answer some frequently asked questions about the function $f$ and its domain.

Q&A

Q: What is the domain of a function?

A: 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 $x$ for which the function $f(x)$ is defined.

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

A: Finding the domain of a function is important because it helps us to determine the values of $x$ for which the function is defined. This is crucial in many mathematical applications, such as solving equations and inequalities, and graphing functions.

Q: How do you find the domain of a function?

A: To find the domain of a function, we need to find the values of $x$ that make the denominator of the function equal to zero. This is because division by zero is undefined, and therefore, the function is not defined at those values of $x$.

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

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

• Not considering the values of $x$ that make the denominator equal to zero
• Not checking for any other restrictions on the domain, such as values of $x$ that make the numerator equal to zero
• Not considering the domain of the function in the context of the problem or application

Q: Can you give an example of a function with a restricted domain?

A: Yes, consider the function $f(x) = \frac{1}{x}$. The domain of this function is all real numbers except $x = 0$, because division by zero is undefined.

Q: How do you graph a function with a restricted domain?

A: To graph a function with a restricted domain, we need to identify the values of $x$ that are not in the domain of the function and exclude them from the graph. We can do this by plotting the function on a coordinate plane and using a dashed line or a dotted line to indicate the restricted domain.

Q: Can you give an example of a function with a restricted domain and its graph?

A: Yes, consider the function $f(x) = \frac{x^2 - 12x + 32}{x^2 - 7x - 8}$. The domain of this function is all real numbers except $x = -1$ and $x = 8$, because these values of $x$ make the denominator equal to zero. The graph of this function would be a dashed line or a dotted line, with the values of $x = -1$ and $x = 8$ excluded from the graph.

Conclusion

In conclusion, finding the domain of a function is an important step in understanding the behavior of the function. By identifying the values of $x$ that are not in the domain of the function, we can determine the restrictions on the domain and graph the function accordingly.

Final Answer

The final answer is $\boxed{-1, 8}$.