The Function \[$ H \$\] Is Defined By $ H(x) = \frac{3x + 2}{x + 3} $.Find $ H(x + 3) $.$ H(x + 3) = \square $

Introduction

In mathematics, functions play a crucial role in modeling real-world phenomena. A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). In this article, we will explore the function $h(x) = \frac{3x + 2}{x + 3}$ and find its transformation when the input is shifted by 3 units.

The Original Function

The function $h(x) = \frac{3x + 2}{x + 3}$ is a rational function, which means it is the ratio of two polynomials. The numerator is $3x + 2$, and the denominator is $x + 3$. To understand the behavior of this function, let's analyze its components.

• The numerator $3x + 2$ is a linear function that increases as $x$ increases.
• The denominator $x + 3$ is also a linear function that increases as $x$ increases.

Finding h(x + 3)

To find $h(x + 3)$, we need to substitute $(x + 3)$ into the original function $h(x) = \frac{3x + 2}{x + 3}$. This means we will replace every instance of $x$ with $(x + 3)$.

h(x + 3) = \frac{3(x + 3) + 2}{(x + 3) + 3}

Simplifying the Expression

Now, let's simplify the expression by expanding the numerator and denominator.

h(x + 3) = \frac{3x + 9 + 2}{x + 6}

Combine like terms in the numerator:

h(x + 3) = \frac{3x + 11}{x + 6}

Final Answer

Therefore, the value of $h(x + 3)$ is:

h(x + 3) = \frac{3x + 11}{x + 6}

Conclusion

In this article, we defined the function $h(x) = \frac{3x + 2}{x + 3}$ and found its transformation when the input is shifted by 3 units. We simplified the expression to obtain the final answer: $h(x + 3) = \frac{3x + 11}{x + 6}$. This result demonstrates the importance of understanding function transformations in mathematics.

Key Takeaways

• Functions can be transformed by shifting the input.
• The transformation of a function can be found by substituting the new input into the original function.
• Simplifying expressions is an essential step in finding the final answer.

Further Exploration

• Investigate other types of function transformations, such as vertical shifts and horizontal stretches.
• Explore the properties of rational functions, including their domains and ranges.
• Apply function transformations to real-world problems, such as modeling population growth or economic trends.
  The Function h(x) and Its Transformation: Q&A =====================================================

Introduction

In our previous article, we explored the function $h(x) = \frac{3x + 2}{x + 3}$ and found its transformation when the input is shifted by 3 units. In this article, we will answer some frequently asked questions about the function and its transformation.

Q: What is the original function h(x)?

A: The original function $h(x)$ is defined as $h(x) = \frac{3x + 2}{x + 3}$.

Q: How do I find h(x + 3)?

A: To find $h(x + 3)$, substitute $(x + 3)$ into the original function $h(x) = \frac{3x + 2}{x + 3}$. This means replacing every instance of $x$ with $(x + 3)$.

Q: What is the simplified expression for h(x + 3)?

A: The simplified expression for $h(x + 3)$ is $\frac{3x + 11}{x + 6}$.

Q: What is the final answer for h(x + 3)?

A: The final answer for $h(x + 3)$ is $\frac{3x + 11}{x + 6}$.

Q: Why is it important to understand function transformations?

A: Understanding function transformations is crucial in mathematics because it allows us to model real-world phenomena and make predictions about future events. Function transformations can be used to describe population growth, economic trends, and many other complex systems.

Q: What are some other types of function transformations?

A: Some other types of function transformations include:

• Vertical shifts: Shifting the function up or down by a certain amount.
• Horizontal stretches: Stretching the function horizontally by a certain factor.
• Horizontal compressions: Compressing the function horizontally by a certain factor.

Q: How do I apply function transformations to real-world problems?

A: To apply function transformations to real-world problems, follow these steps:

1. Identify the problem: Determine the type of problem you are trying to solve.
2. Choose a function: Select a function that models the problem.
3. Apply the transformation: Apply the necessary transformation to the function.
4. Simplify the expression: Simplify the expression to obtain the final answer.

Q: What are some common mistakes to avoid when working with function transformations?

A: Some common mistakes to avoid when working with function transformations include:

• Not simplifying the expression: Failing to simplify the expression can lead to incorrect answers.
• Not checking the domain: Failing to check the domain of the function can lead to incorrect answers.
• Not considering the context: Failing to consider the context of the problem can lead to incorrect answers.

Conclusion

In this article, we answered some frequently asked questions about the function $h(x) = \frac{3x + 2}{x + 3}$ and its transformation. We also discussed the importance of understanding function transformations and provided some tips for applying function transformations to real-world problems. By following these tips and avoiding common mistakes, you can become proficient in working with function transformations and apply them to a wide range of problems.

Key Takeaways

• Understanding function transformations is crucial in mathematics.
• Function transformations can be used to model real-world phenomena.
• Simplifying expressions is an essential step in finding the final answer.
• Avoiding common mistakes is crucial when working with function transformations.

Further Exploration

• Investigate other types of function transformations, such as vertical shifts and horizontal stretches.
• Explore the properties of rational functions, including their domains and ranges.
• Apply function transformations to real-world problems, such as modeling population growth or economic trends.