The Functions F F F And G G G Are Defined As Follows: F ( X ) = X + 3 X 2 − 9 G ( X ) = X X 2 + 25 \begin{array}{l} f(x)=\frac{x+3}{x^2-9} \\ g(x)=\frac{x}{x^2+25} \end{array} F ( X ) = X 2 − 9 X + 3 ​ G ( X ) = X 2 + 25 X ​ ​ For Each Function, Find The Domain. Write Each Answer As An Interval Or Union Of

Introduction

In this article, we will explore the functions $f$ and $g$, which are defined as follows:

$$\begin{array}{l} f(x)=\frac{x+3}{x^2-9} \\ g(x)=\frac{x}{x^2+25} \end{array}$$

Our goal is to find the domain of each function, which is the set of all possible input values for which the function is defined. We will use interval notation to represent the domain of each function.

Finding the Domain of f(x)

To find the domain of $f(x)$, we need to consider the values of $x$ that make the denominator $x^2-9$ equal to zero. This is because division by zero is undefined.

Let's set the denominator equal to zero and solve for $x$:

$$x^2-9=0$$

We can factor the left-hand side as follows:

$$(x-3)(x+3)=0$$

This tells us that either $x-3=0$ or $x+3=0$. Solving for $x$ in each case, we get:

$$x-3=0 \Rightarrow x=3$$

$$x+3=0 \Rightarrow x=-3$$

Therefore, the values of $x$ that make the denominator equal to zero are $x=3$ and $x=-3$.

Since the denominator cannot be equal to zero, we must exclude these values from the domain of $f(x)$. This means that the domain of $f(x)$ is all real numbers except $x=3$ and $x=-3$.

In interval notation, we can represent the domain of $f(x)$ as follows:

$$(-\infty, -3) \cup (-3, 3) \cup (3, \infty)$$

Finding the Domain of g(x)

To find the domain of $g(x)$, we need to consider the values of $x$ that make the denominator $x^2+25$ equal to zero. This is because division by zero is undefined.

Let's set the denominator equal to zero and solve for $x$:

$$x^2+25=0$$

We can see that the left-hand side is always positive, since $x^2$ is always non-negative and $25$ is positive. Therefore, the denominator can never be equal to zero.

This means that the domain of $g(x)$ is all real numbers. In interval notation, we can represent the domain of $g(x)$ as follows:

$$(-\infty, \infty)$$

Conclusion

In this article, we found the domain of the functions $f$ and $g$. The domain of $f(x)$ is all real numbers except $x=3$ and $x=-3$, while the domain of $g(x)$ is all real numbers.

We used interval notation to represent the domain of each function, which provides a clear and concise way to describe the set of all possible input values for which the function is defined.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart

Further Reading

For more information on finding the domain of functions, see the following resources:

• [1] Khan Academy: Finding the domain of a function
• [2] Mathway: Finding the domain of a function

Discussion

Introduction

In our previous article, we explored the functions $f$ and $g$, which are defined as follows:

$$\begin{array}{l} f(x)=\frac{x+3}{x^2-9} \\ g(x)=\frac{x}{x^2+25} \end{array}$$

We found the domain of each function, which is the set of all possible input values for which the function is defined. In this article, we will answer some common questions about the functions $f$ and $g$.

Q&A

Q: What is the domain of f(x)?

A: The domain of $f(x)$ is all real numbers except $x=3$ and $x=-3$. This can be represented in interval notation as:

$$(-\infty, -3) \cup (-3, 3) \cup (3, \infty)$$

Q: What is the domain of g(x)?

A: The domain of $g(x)$ is all real numbers. This can be represented in interval notation as:

$$(-\infty, \infty)$$

Q: Why is the domain of f(x) not all real numbers?

A: The domain of $f(x)$ is not all real numbers because the denominator $x^2-9$ can be equal to zero when $x=3$ or $x=-3$. This makes the function undefined at these points.

Q: Why is the domain of g(x) all real numbers?

A: The domain of $g(x)$ is all real numbers because the denominator $x^2+25$ can never be equal to zero. This is because $x^2$ is always non-negative and $25$ is positive, so the sum $x^2+25$ is always positive.

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

A: To find the domain of a function, you need to consider the values of $x$ that make the denominator equal to zero. You can then exclude these values from the domain.

Q: What is interval notation?

A: Interval notation is a way of representing a set of numbers using a specific notation. It is commonly used to represent the domain of a function.

Q: How do I use interval notation to represent the domain of a function?

A: To use interval notation to represent the domain of a function, you need to identify the values of $x$ that are included in the domain and the values of $x$ that are excluded. You can then use the following notation to represent the domain:

• $(a, b)$ represents the interval from $a$ to $b$ (excluding $a$ and $b$)
• $[a, b]$ represents the interval from $a$ to $b$ (including $a$ and $b$)
• $(-\infty, a)$ represents the interval from negative infinity to $a$
• $(a, \infty)$ represents the interval from $a$ to positive infinity

Conclusion

In this article, we answered some common questions about the functions $f$ and $g$. We also provided some tips and resources for finding the domain of a function and using interval notation to represent the domain.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart

Further Reading

For more information on finding the domain of functions and using interval notation, see the following resources:

• [1] Khan Academy: Finding the domain of a function
• [2] Mathway: Finding the domain of a function
• [3] Wolfram MathWorld: Interval notation

Discussion

What are some common mistakes to avoid when finding the domain of a function? How can you use interval notation to represent the domain of a function? Share your thoughts and experiences in the comments below!