What Is The Domain Of $\left(\frac{b}{q}\right)(x$\]?$ \begin{array}{l} b(x) = -x^2 + 6x \\ q(x) = -4x + 2 \end{array} $A. All Real Numbers Except 0 And 6B. All Real Numbers Except $\frac{1}{2}$C. All Real NumbersD. All Real

Understanding the Domain of a Rational Function

In mathematics, a rational function is a function that can be expressed as the ratio of two polynomials. The domain of a rational function is the set of all possible input values (x-values) for which the function is defined. In other words, it is the set of all possible x-values that the function can accept without resulting in an undefined or imaginary value.

The Domain of a Rational Function: A Closer Look

To determine the domain of a rational function, we need to consider the values of x that make the denominator of the function equal to zero. This is because division by zero is undefined in mathematics. Therefore, we need to find the values of x that make the denominator zero and exclude them from the domain.

The Given Rational Function

The given rational function is $\left(\frac{b}{q}\right)(x)$. The numerator and denominator of the function are given by the following polynomials:

$$\begin{array}{l} b(x) = -x^2 + 6x \\ q(x) = -4x + 2 \end{array}$$

Finding the Domain of the Rational Function

To find the domain of the rational function, we need to determine the values of x that make the denominator $q(x)$ equal to zero. We can do this by setting $q(x)$ equal to zero and solving for x.

$$\begin{align*} q(x) &= -4x + 2 \\ 0 &= -4x + 2 \\ 4x &= 2 \\ x &= \frac{2}{4} \\ x &= \frac{1}{2} \end{align*}$$

Therefore, the value of x that makes the denominator $q(x)$ equal to zero is $x = \frac{1}{2}$.

The Domain of the Rational Function

Since the value of x that makes the denominator $q(x)$ equal to zero is $x = \frac{1}{2}$, we need to exclude this value from the domain of the rational function. Therefore, the domain of the rational function is all real numbers except $\frac{1}{2}$.

Conclusion

In conclusion, the domain of a rational function is the set of all possible input values (x-values) for which the function is defined. To determine the domain of a rational function, we need to consider the values of x that make the denominator of the function equal to zero and exclude them from the domain. In this case, the domain of the rational function $\left(\frac{b}{q}\right)(x)$ is all real numbers except $\frac{1}{2}$.

Answer

Frequently Asked Questions

In this article, we will answer some frequently asked questions about the domain of a rational function.

Q: What is the domain of a rational function?

A: The domain of a rational function is the set of all possible input values (x-values) for which the function is defined. In other words, it is the set of all possible x-values that the function can accept without resulting in an undefined or imaginary value.

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

A: To determine the domain of a rational function, you need to consider the values of x that make the denominator of the function equal to zero. This is because division by zero is undefined in mathematics. Therefore, you need to find the values of x that make the denominator zero and exclude them from the domain.

Q: What happens if the denominator of a rational function is zero?

A: If the denominator of a rational function is zero, the function is undefined at that point. This is because division by zero is undefined in mathematics.

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

A: Yes, a rational function can have a domain of all real numbers if the denominator of the function is never equal to zero. This means that the denominator must be a non-zero constant or a polynomial that never equals zero.

Q: Can a rational function have a domain of all real numbers except a certain value?

A: Yes, a rational function can have a domain of all real numbers except a certain value if the denominator of the function equals zero at that value. This means that the denominator must be a polynomial that equals zero at that value.

Q: How do I find the domain of a rational function with a polynomial numerator and denominator?

A: To find the domain of a rational function with a polynomial numerator and denominator, you need to set the denominator equal to zero and solve for x. This will give you the values of x that make the denominator zero, which you need to exclude from the domain.

Q: What is the domain of the rational function $\left(\frac{b}{q}\right)(x)$?

A: The domain of the rational function $\left(\frac{b}{q}\right)(x)$ is all real numbers except $\frac{1}{2}$. This is because the denominator $q(x)$ equals zero when $x = \frac{1}{2}$.

Q: Can I use a calculator to find the domain of a rational function?

A: Yes, you can use a calculator to find the domain of a rational function. However, you need to make sure that the calculator is set to the correct mode and that you are using the correct function.

Q: How do I graph a rational function with a restricted domain?

A: To graph a rational function with a restricted domain, you need to exclude the values of x that make the denominator zero from the graph. This means that you need to create a graph that shows the function as undefined at those points.

Conclusion

In conclusion, the domain of a rational function is the set of all possible input values (x-values) for which the function is defined. To determine the domain of a rational function, you need to consider the values of x that make the denominator of the function equal to zero and exclude them from the domain. We hope that this article has helped you to understand the concept of the domain of a rational function and how to find it.