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

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Hey guys! Today, we're diving deep into the fascinating world of functions, specifically focusing on the quadratic function f(x) = -(x+3)(x-1). We'll explore its graph and, more importantly, pinpoint its domain and range. Understanding these two concepts is crucial for grasping the behavior and characteristics of any function. So, buckle up and let's embark on this mathematical journey together!

What are Domain and Range, Anyway?

Before we jump into the specifics of our function, let's quickly define what domain and range actually mean. Think of a function as a machine that takes in inputs and spits out outputs. The domain is the set of all possible inputs that the machine can accept without causing any errors. In simpler terms, it's all the x-values for which the function is defined. The range, on the other hand, is the set of all possible outputs that the machine can produce. It represents all the y-values that the function can take on.

For example, consider a simple function like f(x) = 1/x. We can't plug in x = 0 because division by zero is undefined. So, 0 is not in the domain of this function. The domain would be all real numbers except 0. The range, in this case, would also be all real numbers except 0 because there's no input that would ever make the function output 0.

Understanding the domain and range helps us to paint a complete picture of the function's behavior. It tells us where the function exists, what values it can take, and what values it cannot.

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

Now, let's turn our attention to our star function: f(x) = -(x+3)(x-1). This is a quadratic function, which means its graph will be a parabola – a U-shaped curve. The negative sign in front of the expression tells us that the parabola will open downwards, like an upside-down U.

To get a better grasp of this function, let's first expand it into its standard quadratic form: f(x) = -x² -2x + 3. This form helps us identify the coefficients and understand the general shape of the parabola.

Finding the Roots

One of the key features of a parabola is its roots, also known as x-intercepts. These are the points where the graph intersects the x-axis, meaning f(x) = 0. To find the roots, we set the function equal to zero and solve for x:

-(x+3)(x-1) = 0

This equation is already factored, which makes our job easier. We have two factors: (x+3) and (x-1). For the product of two factors to be zero, at least one of them must be zero. So, we set each factor equal to zero and solve:

x + 3 = 0 => x = -3 x - 1 = 0 => x = 1

Therefore, the roots of our function are x = -3 and x = 1. This means the parabola crosses the x-axis at the points (-3, 0) and (1, 0).

Locating the Vertex

Another crucial point on the parabola is the vertex. This is the point where the parabola reaches its maximum or minimum value. Since our parabola opens downwards, the vertex will be the highest point on the graph.

There are a couple of ways to find the vertex. One way is to use the formula for the x-coordinate of the vertex: x_vertex = -b / 2a, where a and b are the coefficients of the quadratic function in standard form (f(x) = ax² + bx + c). In our case, a = -1 and b = -2, so:

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

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging it back into the function:

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

So, the vertex of our parabola is at the point (-1, 4).

Unveiling the Domain of f(x) = -(x+3)(x-1)

Now that we have a good understanding of the function's graph, let's tackle the domain. Remember, the domain is the set of all possible x-values for which the function is defined.

For quadratic functions, there are generally no restrictions on the input values. We can plug in any real number for x, and the function will produce a valid output. There are no denominators that could be zero, no square roots of negative numbers, or any other restrictions that would limit the domain.

Therefore, the domain of f(x) = -(x+3)(x-1) is all real numbers. We can express this mathematically as: (-∞, ∞), which means x can be any number from negative infinity to positive infinity. This is a key characteristic of polynomial functions; they are defined for all real numbers.

Delving into the Range of f(x) = -(x+3)(x-1)

The range, as we discussed, is the set of all possible y-values that the function can take. Since our parabola opens downwards and its vertex is at (-1, 4), the maximum y-value the function can reach is 4. The parabola extends downwards indefinitely, meaning the y-values can go as low as negative infinity.

Therefore, the range of f(x) = -(x+3)(x-1) is all real numbers less than or equal to 4. We can express this mathematically as: (-∞, 4]. The square bracket indicates that 4 is included in the range.

Visualizing the Range on the Graph

Looking at the graph, you can clearly see that the highest point is indeed at y = 4. The graph then stretches downwards, covering all y-values below 4. This visual confirmation reinforces our understanding of the range.

Analyzing the Given Options

Now that we've determined the domain and range of the function, let's look at the options provided and see which one is correct.

A. The domain is all real numbers less than or equal to 4, and the range is all real numbers such that -3 ≤ x ≤ 1.

This option is incorrect. We've established that the domain is all real numbers, not just those less than or equal to 4. Also, the range is all real numbers less than or equal to 4, not a restricted interval between -3 and 1. The values -3 and 1 are the roots (x-intercepts), not the limits of the range.

Given the analysis above, we can confidently say that Option A is incorrect. The domain is all real numbers, and the range is all real numbers less than or equal to 4.

Key Takeaways

  • The domain of a function is the set of all possible input values (x-values). For quadratic functions, the domain is usually all real numbers.
  • The range of a function is the set of all possible output values (y-values).
  • For a downward-opening parabola, the range is all real numbers less than or equal to the y-coordinate of the vertex.
  • Finding the roots and vertex is crucial for understanding the shape and behavior of a parabola.

Final Thoughts

Understanding the domain and range is a fundamental concept in mathematics, especially when dealing with functions. By analyzing the equation and the graph of f(x) = -(x+3)(x-1), we were able to accurately determine its domain and range. Remember, practice makes perfect! So, keep exploring different functions and honing your skills. You've got this!