What Are The Domain And Range Of The Function F ( X ) = − 3 ( X − 5 ) 2 + 4 F(x) = -3(x-5)^2 + 4 F ( X ) = − 3 ( X − 5 ) 2 + 4 ?A. Domain: $(-\infty, 5], Range: ( − ∞ , ∞ (-\infty, \infty ( − ∞ , ∞ ]B. Domain: $(-\infty, 4], Range: ( − ∞ , ∞ (-\infty, \infty ( − ∞ , ∞ ]C. Domain: $(-\infty,

Introduction

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. The domain of a function is the set of all possible input values, while the range is the set of all possible output values. In this article, we will explore the domain and range of the quadratic function $f(x) = -3(x-5)^2 + 4$.

What is a Quadratic Function?

A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The graph of a quadratic function is a parabola, which is a U-shaped curve.

The Given Function

The given function is $f(x) = -3(x-5)^2 + 4$. This function can be rewritten as $f(x) = -3(x^2 - 10x + 25) + 4$, using the formula $(a-b)^2 = a^2 - 2ab + b^2$. Simplifying further, we get $f(x) = -3x^2 + 30x - 75 + 4$, which can be written as $f(x) = -3x^2 + 30x - 71$.

Domain of the Function

The domain of a function is the set of all possible input values. In the case of the quadratic function $f(x) = -3(x-5)^2 + 4$, the domain is all real numbers, since there are no restrictions on the input values. Therefore, the domain of the function is $(-\infty, \infty)$.

Range of the Function

The range of a function is the set of all possible output values. To find the range of the function $f(x) = -3(x-5)^2 + 4$, we need to find the minimum and maximum values of the function. Since the function is a quadratic function, its graph is a parabola that opens downwards, which means the minimum value of the function is the vertex of the parabola.

Finding the Vertex of the Parabola

The vertex of a parabola in the form $f(x) = ax^2 + bx + c$ is given by the formula $x = -\frac{b}{2a}$. In the case of the function $f(x) = -3(x-5)^2 + 4$, we have $a = -3$ and $b = 30$. Plugging these values into the formula, we get $x = -\frac{30}{2(-3)} = 5$. Therefore, the vertex of the parabola is at $x = 5$.

Finding the Minimum Value of the Function

To find the minimum value of the function, we need to plug the value of $x$ into the function. Plugging $x = 5$ into the function $f(x) = -3(x-5)^2 + 4$, we get $f(5) = -3(5-5)^2 + 4 = -3(0)^2 + 4 = 4$. Therefore, the minimum value of the function is $4$.

Finding the Maximum Value of the Function

Since the function is a quadratic function that opens downwards, the maximum value of the function is not defined. However, we can find the maximum value of the function by finding the limit of the function as $x$ approaches infinity. As $x$ approaches infinity, the value of the function approaches negative infinity. Therefore, the maximum value of the function is not defined.

Conclusion

In conclusion, the domain of the function $f(x) = -3(x-5)^2 + 4$ is $(-\infty, \infty)$, and the range of the function is $(-\infty, 4]$. Therefore, the correct answer is:

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

Final Answer

Introduction

In our previous article, we explored the domain and range of the quadratic function $f(x) = -3(x-5)^2 + 4$. In this article, we will answer some frequently asked questions about the domain and range of quadratic functions.

Q: What is the domain of a quadratic function?

A: The domain of a quadratic function is the set of all possible input values. In the case of a quadratic function in the form $f(x) = ax^2 + bx + c$, the domain is all real numbers, since there are no restrictions on the input values.

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

A: To find the domain of a quadratic function, you simply need to look at the function and see if there are any restrictions on the input values. If there are no restrictions, then the domain is all real numbers.

Q: What is the range of a quadratic function?

A: The range of a quadratic function is the set of all possible output values. In the case of a quadratic function in the form $f(x) = ax^2 + bx + c$, the range is all real numbers, since the function can take on any real value.

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

A: To find the range of a quadratic function, you need to find the minimum and maximum values of the function. The minimum value of the function is the vertex of the parabola, and the maximum value of the function is not defined, since the function can take on any real value.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point on the parabola where the function changes from decreasing to increasing, or from increasing to decreasing. The vertex is given by the formula $x = -\frac{b}{2a}$.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you need to plug the values of $a$ and $b$ into the formula $x = -\frac{b}{2a}$. This will give you the x-coordinate of the vertex.

Q: What is the minimum value of a quadratic function?

A: The minimum value of a quadratic function is the value of the function at the vertex of the parabola. This is the lowest point on the parabola.

Q: How do I find the minimum value of a quadratic function?

A: To find the minimum value of a quadratic function, you need to plug the x-coordinate of the vertex into the function. This will give you the minimum value of the function.

Q: What is the maximum value of a quadratic function?

A: The maximum value of a quadratic function is not defined, since the function can take on any real value.

Q: How do I find the maximum value of a quadratic function?

A: To find the maximum value of a quadratic function, you need to find the limit of the function as $x$ approaches infinity. As $x$ approaches infinity, the value of the function approaches negative infinity.

Conclusion

In conclusion, the domain and range of a quadratic function are all real numbers. The vertex of a parabola is the point on the parabola where the function changes from decreasing to increasing, or from increasing to decreasing. The minimum value of a quadratic function is the value of the function at the vertex of the parabola, and the maximum value of a quadratic function is not defined.

Final Answer

The final answer is that the domain and range of a quadratic function are all real numbers, and the vertex of a parabola is the point on the parabola where the function changes from decreasing to increasing, or from increasing to decreasing.