Find The Domain Of The Function $f(x)=\frac{1}{2x+5}$. What Is The Only Value Of $x$ Not In The Domain?Only Value = $\square$

Feb 28, 2025 by ADMIN 132 views

**Domain of a Function: Understanding the Restrictions**

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 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 the function $f(x)=\frac{1}{2x+5}$ and identifying the only value of $x$ that is not in the domain.

What is the Domain of a Function?

The domain of a function is a crucial concept in mathematics, as it helps us understand the restrictions on the input values that the function can accept. A function is defined as 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.

Restrictions on the Domain

There are several restrictions on the domain of a function, including:

Division by zero: A function is undefined when the denominator is equal to zero.
Square root of a negative number: A function is undefined when the expression inside the square root is negative.
Logarithm of a non-positive number: A function is undefined when the argument of the logarithm is less than or equal to zero.

Finding the Domain of the Function

To find the domain of the function $f(x)=\frac{1}{2x+5}$, we need to identify the values of $x$ that make the denominator equal to zero. The denominator is equal to zero when $2x+5=0$.

Solving the Equation

To solve the equation $2x+5=0$, we need to isolate the variable $x$. We can do this by subtracting 5 from both sides of the equation and then dividing both sides by 2.

2x+5=0

2x=-5

x=-\frac{5}{2}

Identifying the Value of x Not in the Domain

The value of $x$ that makes the denominator equal to zero is $x=-\frac{5}{2}$. This is the only value of $x$ that is not in the domain of the function.

Conclusion

In conclusion, the domain of the function $f(x)=\frac{1}{2x+5}$ is all real numbers except $x=-\frac{5}{2}$. This is because the function is undefined when the denominator is equal to zero, which occurs when $x=-\frac{5}{2}$. Therefore, the only value of $x$ not in the domain is $x=-\frac{5}{2}$.

Final Answer

The only value of $x$ not in the domain is $x=-\frac{5}{2}$.

Example Use Case

Suppose we want to find the value of the function $f(x)=\frac{1}{2x+5}$ when $x=0$. We can plug in $x=0$ into the function and simplify:

f(0)=\frac{1}{2(0)+5}

f(0)=\frac{1}{5}

This shows that the function is defined when $x=0$, which is a value in the domain of the function.

Tips and Tricks

When finding the domain of a function, always check for division by zero and square roots of negative numbers.
Use algebraic manipulations to simplify the expression and identify the values of the variable that make the denominator equal to zero.
Be careful when plugging in values into the function, as this can help you identify the values of the variable that are in the domain.

Common Mistakes

Failing to check for division by zero and square roots of negative numbers.
Not simplifying the expression to identify the values of the variable that make the denominator equal to zero.
Plugging in values into the function without checking if they are in the domain.

Real-World Applications

Finding the domain of a function is crucial in many real-world applications, such as economics, physics, and engineering.
Understanding the domain of a function can help us make informed decisions and predictions about the behavior of the function.
The domain of a function can also help us identify the values of the variable that are relevant to the problem at hand.

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 the variable that the function can accept without resulting in an undefined or imaginary output.

Q: What are the restrictions on the domain of a function?

A: There are several restrictions on the domain of a function, including:

Division by zero: A function is undefined when the denominator is equal to zero.
Square root of a negative number: A function is undefined when the expression inside the square root is negative.
Logarithm of a non-positive number: A function is undefined when the argument of the logarithm is less than or equal to zero.

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

A: To find the domain of a function, you need to identify the values of the variable that make the denominator equal to zero. You can do this by solving the equation obtained by setting the denominator equal to zero.

Q: What is the only value of x not in the domain of the function f(x)=1/(2x+5)?

A: The only value of x not in the domain of the function f(x)=1/(2x+5) is x=-5/2.

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

A: Yes, consider the function f(x)=1/x. The domain of this function is all real numbers except x=0, because the function is undefined when the denominator is equal to zero.

Q: How do I know if a function is defined at a particular value of x?

A: To determine if a function is defined at a particular value of x, you need to check if the denominator is equal to zero at that value of x. If the denominator is not equal to zero, then the function is defined at that value of x.

Q: Can you explain the concept of domain in the context of real-world applications?

A: Yes, the concept of domain is crucial in many real-world applications, such as economics, physics, and engineering. For example, in economics, the domain of a function can represent the set of all possible input values for a particular economic model. In physics, the domain of a function can represent the set of all possible input values for a particular physical system.

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:

Failing to check for division by zero and square roots of negative numbers.
Not simplifying the expression to identify the values of the variable that make the denominator equal to zero.
Plugging in values into the function without checking if they are in the domain.

Q: Can you provide some tips and tricks for finding the domain of a function?

A: Yes, here are some tips and tricks for finding the domain of a function:

Always check for division by zero and square roots of negative numbers.
Simplify the expression to identify the values of the variable that make the denominator equal to zero.
Be careful when plugging in values into the function, as this can help you identify the values of the variable that are in the domain.

Q: What are some real-world applications of the concept of domain?

A: Some real-world applications of the concept of domain include:

Economics: The domain of a function can represent the set of all possible input values for a particular economic model.
Physics: The domain of a function can represent the set of all possible input values for a particular physical system.
Engineering: The domain of a function can represent the set of all possible input values for a particular engineering system.

Q: Can you recommend some resources for further learning on the concept of domain?

A: Yes, here are some resources for further learning on the concept of domain:

Wikipedia: Domain of a Function
MathWorld: Domain of a Function
Calculus by Michael Spivak: Chapter 1, Section 1.4

Q: What is the significance of the domain of a function in mathematics?

A: The domain of a function is a crucial concept in mathematics, as it helps us understand the restrictions on the input values that the function can accept. Understanding the domain of a function is essential for making informed decisions and predictions about the behavior of the function.