Select The Correct Answer.What Are The Domain And Range Of This Function? Y = ( X + 3 ) 2 − 5 Y=(x+3)^2-5 Y = ( X + 3 ) 2 − 5 A. Domain: ( − ∞ , ∞ (-\infty, \infty ( − ∞ , ∞ ] Range: {-5, \infty }$B. Domain: ( − ∞ , ∞ (-\infty, \infty ( − ∞ , ∞ ] Range: ( − ∞ , ∞ (-\infty, \infty ( − ∞ , ∞ ]C.

Introduction

When dealing with functions, it's essential to understand the concept of domain and range. The domain of a function is the set of all possible input values (x-values) that the function can accept, while the range is the set of all possible output values (y-values) that the function can produce. In this article, we will explore the domain and range of the function $y=(x+3)^2-5$.

What is the Domain of a Function?

The domain of a function is the set of all possible input values (x-values) that the function can accept. In other words, it's the set of all possible values of x that will produce a valid output value of y. For a function to be defined, the input value (x) must be a real number.

What is the Range of a Function?

The range of a function is the set of all possible output values (y-values) that the function can produce. In other words, it's the set of all possible values of y that will result from a valid input value of x.

Analyzing the Function $y=(x+3)^2-5$

The given function is $y=(x+3)^2-5$. To find the domain and range of this function, we need to analyze its behavior.

Squaring the Binomial

The function $y=(x+3)^2-5$ involves squaring the binomial $(x+3)$. When we square a binomial, we get a perfect square trinomial. In this case, the perfect square trinomial is $(x+3)^2 = x^2 + 6x + 9$.

Subtracting 5

After squaring the binomial, we subtract 5 from the result. This gives us the function $y = x^2 + 6x + 9 - 5 = x^2 + 6x + 4$.

Analyzing the Function

Now that we have simplified the function, let's analyze its behavior. The function $y = x^2 + 6x + 4$ is a quadratic function, which means it has a parabolic shape. The parabola opens upward because the coefficient of the $x^2$ term is positive.

Finding the Domain

Since the function is a quadratic function, its domain is all real numbers. In other words, the domain of the function $y = x^2 + 6x + 4$ is $(-\infty, \infty)$.

Finding the Range

To find the range of the function, we need to find the minimum value of the function. The minimum value of a quadratic function occurs at its vertex. The x-coordinate of the vertex is given by the formula $x = -\frac{b}{2a}$, where $a$ is the coefficient of the $x^2$ term and $b$ is the coefficient of the $x$ term.

In this case, $a = 1$ and $b = 6$. Plugging these values into the formula, we get $x = -\frac{6}{2(1)} = -3$.

To find the y-coordinate of the vertex, we plug $x = -3$ into the function: $y = (-3)^2 + 6(-3) + 4 = 9 - 18 + 4 = -5$.

Therefore, the minimum value of the function is $y = -5$. Since the parabola opens upward, the range of the function is all values greater than or equal to $-5$. In other words, the range of the function $y = x^2 + 6x + 4$ is $[-5, \infty)$.

Conclusion

In conclusion, the domain of the function $y = x^2 + 6x + 4$ is all real numbers, while the range is all values greater than or equal to $-5$. Therefore, the correct answer is:

A. Domain: $(-\infty, \infty)$ Range: $[-5, \infty)$

Final Answer

Introduction

In our previous article, we discussed the domain and range of the function $y=(x+3)^2-5$. We learned that the domain of a function is the set of all possible input values (x-values) that the function can accept, while the range is the set of all possible output values (y-values) that the function can produce.

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

Q: What is the difference between the domain and range of a function?

A: The domain of a function is the set of all possible input values (x-values) that the function can accept, while the range is the set of all possible output values (y-values) that the function can produce.

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

A: To find the domain of a function, you need to determine the set of all possible input values (x-values) that the function can accept. This can be done by analyzing the function and identifying any restrictions on the input values.

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

A: Some common restrictions on the domain of a function include:

• Division by zero: A function cannot have a denominator of zero, as this would result in an undefined value.
• Square root of a negative number: A function cannot have a square root of a negative number, as this would result in an imaginary value.
• Logarithm of a non-positive number: A function cannot have a logarithm of a non-positive number, as this would result in an undefined value.

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

A: To find the range of a function, you need to determine the set of all possible output values (y-values) that the function can produce. This can be done by analyzing the function and identifying any restrictions on the output values.

Q: What are some common restrictions on the range of a function?

A: Some common restrictions on the range of a function include:

• A function cannot have a range that includes negative values if the function is defined for all real numbers.
• A function cannot have a range that includes positive values if the function is defined for all real numbers.

Q: Can a function have a domain and range that are the same?

A: Yes, a function can have a domain and range that are the same. For example, the function $y=x^2$ has a domain and range of $[0, \infty)$.

Q: Can a function have a domain that is a subset of its range?

A: Yes, a function can have a domain that is a subset of its range. For example, the function $y=x^2$ has a domain of $[0, \infty)$ and a range of $[0, \infty)$.

Q: Can a function have a range that is a subset of its domain?

A: No, a function cannot have a range that is a subset of its domain. This would imply that the function is not defined for all real numbers.

Conclusion

In conclusion, the domain and range of a function are two important concepts that are used to describe the behavior of a function. By understanding the domain and range of a function, you can better analyze and solve problems involving functions.

Final Answer

The final answer is that the domain and range of a function are two important concepts that are used to describe the behavior of a function.

Common Mistakes

• Assuming that the domain and range of a function are the same.
• Assuming that a function can have a range that is a subset of its domain.
• Not analyzing the function to determine the domain and range.

Tips and Tricks

• Always analyze the function to determine the domain and range.
• Use the properties of functions to determine the domain and range.
• Practice solving problems involving functions to improve your understanding of the domain and range.