Find The Domain Of { P $}$ And Write Your Answer In Set Notation And Interval Notation. Round Answers To 3 Decimal Places As Needed. { P(t) = \frac 7}{7t + 5} $}$Set Notation [${ T \mid T \text{ Is A Real Number And T \neq
Finding the Domain of a Rational Function
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 the variable that the function can accept without resulting in an undefined or imaginary output. In this article, we will focus on finding the domain of a rational function, specifically the function { p(t) = \frac{7}{7t + 5} $}$.
Understanding Rational Functions
A rational function is a function that can be expressed as the ratio of two polynomials. In the case of the function { p(t) = \frac{7}{7t + 5} $}$, the numerator is a constant polynomial, and the denominator is a linear polynomial. Rational functions can have various types of restrictions on their domain, including vertical asymptotes, holes, and restrictions on the variable.
Finding the Domain of a Rational Function
To find the domain of a rational function, we need to identify the values of the variable that would result in an undefined or imaginary output. In the case of the function { p(t) = \frac{7}{7t + 5} $}$, the denominator is the only part of the function that can result in an undefined output. Therefore, we need to find the values of { t $}$ that would make the denominator equal to zero.
Solving for the Denominator
To find the values of { t $}$ that would make the denominator equal to zero, we need to set the denominator equal to zero and solve for { t $}$. In this case, we have:
{ 7t + 5 = 0 $}$
Subtracting 5 from both sides of the equation, we get:
{ 7t = -5 $}$
Dividing both sides of the equation by 7, we get:
{ t = -\frac{5}{7} $}$
Therefore, the value of { t $}$ that would make the denominator equal to zero is { -\frac{5}{7} $}$.
Writing the Domain in Set Notation
The domain of a function is typically written in set notation, which is a way of expressing a set of values using mathematical notation. In this case, the domain of the function { p(t) = \frac{7}{7t + 5} $}$ can be written as:
{ { t \mid t \text{ is a real number and } t \neq -\frac{5}{7} } $}$
This notation indicates that the domain of the function is the set of all real numbers except { -\frac{5}{7} $}$.
Writing the Domain in Interval Notation
Interval notation is another way of expressing a set of values using mathematical notation. In this case, the domain of the function { p(t) = \frac{7}{7t + 5} $}$ can be written as:
{ (-\infty, -\frac{5}{7}) \cup (-\frac{5}{7}, \infty) $}$
This notation indicates that the domain of the function is the set of all real numbers except { -\frac{5}{7} $}$.
In conclusion, the domain of the function { p(t) = \frac{7}{7t + 5} $}$ is the set of all real numbers except { -\frac{5}{7} $}$. This can be written in set notation as { { t \mid t \text{ is a real number and } t \neq -\frac{5}{7} } $}$ and in interval notation as { (-\infty, -\frac{5}{7}) \cup (-\frac{5}{7}, \infty) $}$.
Q&A: Finding the Domain of a Rational Function
In our previous article, we discussed how to find the domain of a rational function, specifically the function { p(t) = \frac{7}{7t + 5} $}$. In this article, we will answer some common questions related to finding 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 for which the function is defined. In other words, it is the set of all possible values of the variable that the function can accept without resulting in an undefined or imaginary output.
Q: How do I find the domain of a rational function?
A: To find the domain of a rational function, you need to identify the values of the variable that would result in an undefined or imaginary output. In the case of a rational function, the denominator is the only part of the function that can result in an undefined output. Therefore, you need to find the values of the variable that would make the denominator equal to zero.
Q: What if the denominator is a quadratic expression?
A: If the denominator is a quadratic expression, you need to factor the expression and find the values of the variable that would make the expression equal to zero. You can then use the quadratic formula to find the values of the variable.
Q: What if the denominator is a polynomial of degree 3 or higher?
A: If the denominator is a polynomial of degree 3 or higher, you need to use numerical methods or graphing tools to find the values of the variable that would make the expression equal to zero.
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. Many graphing calculators have a built-in function to find the domain of a rational function.
Q: How do I write the domain of a rational function in set notation?
A: To write the domain of a rational function in set notation, you need to use the following format:
{ { x \mid x \text{ is a real number and } x \neq a } $}$
where { a $}$ is the value of the variable that would make the denominator equal to zero.
Q: How do I write the domain of a rational function in interval notation?
A: To write the domain of a rational function in interval notation, you need to use the following format:
{ (-\infty, a) \cup (a, \infty) $}$
where { a $}$ is the value of the variable that would make the denominator equal to zero.
Q: Can I have multiple values of the variable that would make the denominator equal to zero?
A: Yes, you can have multiple values of the variable that would make the denominator equal to zero. In this case, you need to write the domain of the rational function as a union of intervals.
Q: How do I find the domain of a rational function with multiple variables?
A: To find the domain of a rational function with multiple variables, you need to identify the values of each variable that would result in an undefined or imaginary output. You can then use the same process as before to find the domain of the rational function.
In conclusion, finding the domain of a rational function is an important step in understanding the behavior of the function. By following the steps outlined in this article, you can find the domain of a rational function and write it in set notation and interval notation.