Answer :
- The domain of the quadratic function $f(x) = -x^2 - 18x - 84$ is all real numbers: $(-\infty, \infty)$.
- The x-coordinate of the vertex is calculated as $x = -\frac{b}{2a} = -9$.
- The y-coordinate of the vertex is calculated as $f(-9) = -3$.
- The range of the function, since the parabola opens downwards, is $(-\infty, -3]$.
$\boxed{(-\infty, -3]}$
### Explanation
1. Analyzing the Problem
We are given the quadratic function $f(x) = -x^2 - 18x - 84$. Our goal is to find the domain and range of this function. Since it is a polynomial, we know a few things right off the bat. Let's start with the domain.
2. Determining the Domain
The domain of any polynomial function, including quadratic functions, is all real numbers. This is because we can plug in any real number for $x$ and get a real number output for $f(x)$. In interval notation, this is written as $(-\infty, \infty)$.
3. Understanding the Range
Now, let's find the range. Since the coefficient of the $x^2$ term is negative ($-1$), the parabola opens downwards. This means the function has a maximum value at its vertex. The range will be all real numbers less than or equal to the y-coordinate of the vertex, i.e., $(-\infty, y]$, where $y$ is the y-coordinate of the vertex.
4. Finding the x-coordinate of the Vertex
To find the vertex, we use the formula $x = -\frac{b}{2a}$, where $a = -1$ and $b = -18$. Plugging in these values, we get:
$$x = -\frac{-18}{2(-1)} = -\frac{-18}{-2} = -9$$
5. Finding the y-coordinate of the Vertex
Now we need to find the y-coordinate of the vertex by plugging $x = -9$ into the function:
$$f(-9) = -(-9)^2 - 18(-9) - 84 = -81 + 162 - 84 = -3$$
So, the vertex of the parabola is at $(-9, -3)$.
6. Determining the Range
Since the parabola opens downwards, the range is all real numbers less than or equal to $-3$. In interval notation, this is $(-\infty, -3]$.
7. Final Answer
Therefore, the domain of $f(x)$ is $(-\infty, \infty)$ and the range is $(-\infty, -3]$.
### Examples
Understanding the domain and range of a function is crucial in many real-world applications. For example, if $f(x)$ represents the profit of a company based on the number of items produced ($x$), the domain tells us the possible number of items that can be produced (which can't be negative, so the domain would be restricted). The range tells us the possible profit values the company can achieve. Knowing these bounds helps in making informed business decisions.
- The x-coordinate of the vertex is calculated as $x = -\frac{b}{2a} = -9$.
- The y-coordinate of the vertex is calculated as $f(-9) = -3$.
- The range of the function, since the parabola opens downwards, is $(-\infty, -3]$.
$\boxed{(-\infty, -3]}$
### Explanation
1. Analyzing the Problem
We are given the quadratic function $f(x) = -x^2 - 18x - 84$. Our goal is to find the domain and range of this function. Since it is a polynomial, we know a few things right off the bat. Let's start with the domain.
2. Determining the Domain
The domain of any polynomial function, including quadratic functions, is all real numbers. This is because we can plug in any real number for $x$ and get a real number output for $f(x)$. In interval notation, this is written as $(-\infty, \infty)$.
3. Understanding the Range
Now, let's find the range. Since the coefficient of the $x^2$ term is negative ($-1$), the parabola opens downwards. This means the function has a maximum value at its vertex. The range will be all real numbers less than or equal to the y-coordinate of the vertex, i.e., $(-\infty, y]$, where $y$ is the y-coordinate of the vertex.
4. Finding the x-coordinate of the Vertex
To find the vertex, we use the formula $x = -\frac{b}{2a}$, where $a = -1$ and $b = -18$. Plugging in these values, we get:
$$x = -\frac{-18}{2(-1)} = -\frac{-18}{-2} = -9$$
5. Finding the y-coordinate of the Vertex
Now we need to find the y-coordinate of the vertex by plugging $x = -9$ into the function:
$$f(-9) = -(-9)^2 - 18(-9) - 84 = -81 + 162 - 84 = -3$$
So, the vertex of the parabola is at $(-9, -3)$.
6. Determining the Range
Since the parabola opens downwards, the range is all real numbers less than or equal to $-3$. In interval notation, this is $(-\infty, -3]$.
7. Final Answer
Therefore, the domain of $f(x)$ is $(-\infty, \infty)$ and the range is $(-\infty, -3]$.
### Examples
Understanding the domain and range of a function is crucial in many real-world applications. For example, if $f(x)$ represents the profit of a company based on the number of items produced ($x$), the domain tells us the possible number of items that can be produced (which can't be negative, so the domain would be restricted). The range tells us the possible profit values the company can achieve. Knowing these bounds helps in making informed business decisions.
