Inverse Function Of G(x) = X² - 1 On Positive Real Numbers

Hey guys! Today, we're diving deep into the fascinating world of functions, specifically focusing on the function g(x) = x² - 1. But there's a twist! We're not looking at this function over the entire set of real numbers. Instead, we're restricting our domain to the set of positive real numbers. This seemingly small change opens up a whole new realm of possibilities, especially when we start talking about inverse functions.

Understanding the Function g(x) = x² - 1

Let's first get a solid grasp on the function itself. g(x) = x² - 1 is a quadratic function, meaning its graph is a parabola. If we were to graph it over all real numbers, we'd see a U-shaped curve that opens upwards, with its vertex (the lowest point) at (0, -1). However, because we're only considering positive real numbers as our input (our domain), we're essentially chopping off the left half of the parabola. We're only interested in the part of the curve that lies to the right of the y-axis. This restriction is crucial because it impacts whether the function has an inverse.

When we talk about a function's domain, we're talking about all the possible input values (x-values) that we can plug into the function. In this case, our domain is the set of positive real numbers, which we can write as (0, ∞). This means we can plug in any positive number into g(x), but we can't plug in zero or any negative numbers. The range of a function, on the other hand, is the set of all possible output values (y-values) that the function can produce. For g(x) = x² - 1 with a domain of positive real numbers, the range is (-1, ∞). Think about it: when x is a very small positive number close to zero, x² is also a very small positive number, and x² - 1 will be slightly less than -1. As x gets larger and larger, x² gets much larger, and so does x² - 1. This means our function will produce y-values greater than -1, increasing towards infinity.

Why is restricting the domain important when considering inverses? The key concept here is the horizontal line test. A function has an inverse if and only if it passes the horizontal line test, meaning no horizontal line intersects the graph of the function more than once. If we considered the entire parabola of g(x) = x² - 1, it would fail the horizontal line test miserably. A horizontal line like y = 0 would intersect the parabola twice. But by restricting our domain to positive real numbers, we get rid of the left half of the parabola. Now, our remaining curve passes the horizontal line test with flying colors! This tells us that the restricted version of g(x) does indeed have an inverse.

The Concept of Inverse Functions

Now that we know g(x) has an inverse over the positive real numbers, let's take a step back and clarify what an inverse function actually is. In simple terms, an inverse function