Graphing F(x) = X² - 1: Finding Domain And Range

Hey guys! Let's dive into the fascinating world of functions and graphs, specifically focusing on the function f(x) = x² - 1. We're going to break down how to graph this function and then figure out its domain and range. Trust me, it's not as scary as it sounds! We'll use a casual, friendly approach to make sure everyone understands. So, grab your pencils (or your favorite graphing software) and let's get started!

Graphing the Function f(x) = x² - 1

To truly understand a function, visualizing it is key. When we're talking about f(x) = x² - 1, we're dealing with a quadratic function, which means its graph will be a parabola – a U-shaped curve. This specific function is a transformation of the basic f(x) = x² parabola. Understanding these transformations is fundamental to grasping the behavior of various functions in mathematics. The - 1 in our equation is the game-changer here. It tells us that the entire parabola has been shifted down by 1 unit on the y-axis. Think of it like the parabola having a little slide downwards!

To accurately sketch the graph, we need to plot a few key points. A great starting point is to find the vertex of the parabola. The vertex is the lowest (or highest, depending on the function) point on the curve. For f(x) = x² - 1, the vertex is at the point (0, -1). This is because when x is 0, f(x) is -1, and since x² is always non-negative, this is the minimum value of the function. Plotting this point gives us a solid anchor for our graph. Next, we can choose a few x-values on either side of the vertex and calculate their corresponding f(x) values. For instance, if we plug in x = 1, we get f(1) = 1² - 1 = 0, giving us the point (1, 0). Similarly, if we plug in x = -1, we get f(-1) = (-1)² - 1 = 0, giving us the point (-1, 0). These points are crucial because they help us define the shape and direction of the parabola. By connecting these points with a smooth curve, we can visualize the parabolic shape of our function.

Continuing this process, we can plot more points to get an even clearer picture. For x = 2, f(2) = 2² - 1 = 3, resulting in the point (2, 3). And for x = -2, f(-2) = (-2)² - 1 = 3, resulting in the point (-2, 3). Notice the symmetry here? Parabolas are symmetrical about their vertex, which means for every point on one side of the vertex, there's a corresponding point on the other side. This symmetry is a key characteristic of quadratic functions and makes graphing them much easier. By plotting these additional points, we can confidently draw the parabola, showcasing its U-shape and its position on the coordinate plane. Remember, the more points you plot, the more accurate your graph will be, allowing for a better understanding of the function's behavior. Understanding the graph of f(x) = x² - 1 not only helps in visualizing the function but also lays the foundation for comprehending its domain and range, which we will explore in the subsequent sections. Visual aids like graphs are incredibly powerful tools in mathematics, making abstract concepts more concrete and accessible.

Determining the Domain of f(x) = x² - 1

Alright, let's talk about the domain! In simple terms, the domain of a function is the set of all possible input values (x-values) that the function can accept without causing any mathematical mayhem. Think of it as the function's comfort zone – the values it can handle without throwing an error. For polynomial functions, like our f(x) = x² - 1, the domain is usually pretty straightforward. The domain represents the scope of possible input values for our function, and recognizing any restrictions is critical for accurately interpreting its behavior. In this context, we examine the types of numbers that can be legally input into our function without resulting in undefined outcomes. This is a fundamental concept in understanding how functions operate and is crucial for a range of mathematical applications.

Now, ask yourself: are there any x-values that would make our function explode or give us an undefined result? For f(x) = x² - 1, the answer is a resounding no! We can square any real number, and we can subtract 1 from any real number. There are no restrictions here. We don't have any fractions with x in the denominator (which would be a problem if the denominator became zero), and we don't have any square roots or other even roots (which would be a problem if we tried to take the root of a negative number). This absence of restrictions is a hallmark of polynomial functions, making the determination of their domain relatively simple. Because we can input any real number into f(x) = x² - 1, we say that the domain is all real numbers. This includes everything from negative infinity to positive infinity. Whether you're working with whole numbers, fractions, decimals, or even irrational numbers like pi, you can plug them into this function without any issues. This characteristic is crucial in many mathematical contexts, allowing for smooth and continuous calculations across a wide range of applications.

We can express this mathematically in a couple of ways. One way is to use interval notation, where we write the domain as (-∞, ∞). The parentheses indicate that we're not actually including negative infinity or positive infinity (because infinity isn't a number, it's a concept). Another way to express the domain is using set-builder notation, where we write {x | x ∈ ℝ}. This reads as