Domain & Range Of F(x) = -(x+3)(x-1) Explained

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Hey guys! Let's break down the domain and range of the quadratic function f(x) = -(x+3)(x-1). We've got the graph right here, and we need to figure out what's true about its domain and range. It's a classic algebra problem, and once you get the hang of it, you'll be solving these in your sleep!

Delving into Domain and Range

So, what exactly are domain and range? Think of the domain as all the possible x-values you can plug into your function without breaking any mathematical rules. The most common things that cause problems are dividing by zero (which is a big no-no!) and taking the square root of a negative number (in the realm of real numbers, anyway). Now, the range is all the possible y-values (or f(x) values) that you can get out of the function after plugging in all those valid x-values. Basically, domain is the input, and range is the output.

For polynomial functions, like our quadratic f(x) = -(x+3)(x-1), the domain is generally all real numbers. Why? Because you can plug in any real number for x, and the function will happily spit out a real number result. There are no denominators to worry about, no square roots of potentially negative numbers, nothing like that. Quadratics are super friendly in this way.

But the range is a bit trickier. Since our function is a quadratic, its graph is a parabola. Parabolas either open upwards or downwards, and this affects the range. If the parabola opens upwards, it has a minimum value, and the range is all real numbers greater than or equal to that minimum. If it opens downwards, it has a maximum value, and the range is all real numbers less than or equal to that maximum. Our job is to figure out which way our parabola opens and what that maximum or minimum value is. In this case, because of the negative sign in front of the function, we know the parabola opens downward. Thus, we have to find the maximum point.

Analyzing the Function f(x) = -(x+3)(x-1)

Let's rewrite our function f(x) = -(x+3)(x-1) in standard form to get a better handle on it. Expanding the expression, we get:

f(x) = -(x² + 3x - x - 3) f(x) = -(x² + 2x - 3) f(x) = -x² - 2x + 3

Now we have a quadratic in the form f(x) = ax² + bx + c, where a = -1, b = -2, and c = 3. Because a is negative, the parabola opens downward, confirming what we suspected. The vertex of the parabola represents the maximum point of the function.

The x-coordinate of the vertex can be found using the formula x = -b / 2a. Plugging in our values for a and b, we get:

x = -(-2) / (2 * -1) = 2 / -2 = -1

So, the x-coordinate of the vertex is -1. To find the y-coordinate (which is the maximum value of the function), we plug this x-value back into our function:

f(-1) = -(-1)² - 2(-1) + 3 f(-1) = -1 + 2 + 3 f(-1) = 4

Therefore, the vertex of our parabola is at the point (-1, 4). This means the maximum value of the function is 4. Since the parabola opens downwards, the range of the function is all real numbers less than or equal to 4.

Putting It All Together

Okay, let's recap. The domain of the function f(x) = -(x+3)(x-1) is all real numbers because it's a polynomial function. The range is all real numbers less than or equal to 4 because the parabola opens downwards and has a maximum value of 4 at its vertex. Therefore, the correct answer is:

A. The domain is all real numbers, and the range is all real numbers less than or equal to 4.

Additional Insights on Quadratic Functions

Quadratic functions are a cornerstone of algebra, and mastering their domain and range is super important. Let's explore some additional insights to deepen your understanding.

Understanding the Vertex Form

The vertex form of a quadratic function is given by: f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form makes it incredibly easy to identify the vertex and, consequently, the maximum or minimum value of the function. By completing the square, any quadratic function in the standard form can be converted into vertex form.

Axis of Symmetry

Every parabola has an axis of symmetry, which is a vertical line that passes through the vertex. The equation of the axis of symmetry is x = h, where h is the x-coordinate of the vertex. The parabola is symmetric about this line, meaning that if you were to fold the graph along the axis of symmetry, the two halves would perfectly match.

Intercepts

The x-intercepts are the points where the parabola intersects the x-axis. These points are also known as the roots or zeros of the function. To find the x-intercepts, you set f(x) = 0 and solve for x. For our function f(x) = -(x+3)(x-1), the x-intercepts are x = -3 and x = 1. The y-intercept is the point where the parabola intersects the y-axis. To find the y-intercept, you set x = 0 and evaluate f(0). For our function, f(0) = -(0+3)(0-1) = -(-3) = 3, so the y-intercept is at (0, 3).

The Discriminant

The discriminant of a quadratic equation ax² + bx + c = 0 is given by Δ = b² - 4ac. The discriminant tells you about the nature of the roots of the equation:

  • If Δ > 0, the equation has two distinct real roots (two x-intercepts).
  • If Δ = 0, the equation has one real root (the vertex touches the x-axis).
  • If Δ < 0, the equation has no real roots (the parabola does not intersect the x-axis).

Practical Applications of Quadratic Functions

Quadratic functions aren't just abstract mathematical concepts; they have tons of real-world applications. Here are a few examples:

  • Projectile Motion: The path of a projectile (like a ball thrown in the air) can be modeled by a quadratic function. The vertex of the parabola represents the maximum height reached by the projectile.
  • Optimization Problems: Many optimization problems, where you want to maximize or minimize a certain quantity, can be solved using quadratic functions. For example, finding the dimensions of a rectangular garden that maximize its area given a fixed perimeter.
  • Engineering: Quadratic functions are used in engineering to design bridges, arches, and other structures. The parabolic shape is often used because it distributes weight evenly and provides stability.

By understanding the domain and range of quadratic functions and their various properties, you'll be well-equipped to tackle a wide range of mathematical and real-world problems. Keep practicing, and you'll become a quadratic function pro in no time!