Inverse Of G(x) = X^2 - 2: Is It A Function?

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Let's dive into whether the inverse of the function g(x) = x² - 2 is actually a function itself. We'll break down the process, look at the graph, and use a little bit of math logic to figure it out. So, grab your thinking caps, guys, because here we go!

Understanding the Original Function: g(x) = x² - 2

First, let's get comfy with our original function, g(x) = x² - 2. This is a simple quadratic function. If you were to graph it, you'd see a parabola that opens upwards. The lowest point of the parabola (the vertex) is at (0, -2). Basically, it’s your standard x² parabola, but shifted down by 2 units on the y-axis.

Quadratic functions like this one have a unique property: for every positive y-value (above -2 in our case), there are two corresponding x-values. For instance, if you wanted to find what x-values give you g(x) = 2, you'd solve the equation:

x² - 2 = 2

x² = 4

x = ±2

So, both x = 2 and x = -2 give you a y-value of 2. This little fact is super important when we start thinking about inverses.

Finding the Inverse of g(x)

Okay, now let's find the inverse of g(x). To do this, we're going to swap x and y and then solve for y. Remember, g(x) is just another way of writing y, so we start with:

y = x² - 2

Swap x and y:

x = y² - 2

Now, solve for y:

x + 2 = y²

y = ±√(x + 2)

So, the inverse of g(x) is y = ±√(x + 2). Notice that plus-or-minus sign? That's a huge clue about whether this inverse is a function!

Visualizing the Inverse: Reflection Across the Line y = x

The graph of the inverse of g(x) is the reflection of the graph of g(x) across the line y = x. Imagine you have the graph of g(x) = x² - 2 drawn on a piece of paper. Now, fold the paper along the line y = x. The image you see through the paper is the graph of the inverse. This reflection swaps the x and y coordinates, which is exactly what we did algebraically when we found the inverse.

This reflection visually demonstrates why the inverse might not be a function. Because the original parabola opens upwards, its reflection opens to the right, creating a sideways parabola. This shape immediately raises a red flag because it might fail the vertical line test.

The Vertical Line Test and Why the Inverse Fails

The vertical line test is a simple way to check if a graph represents a function. If you can draw any vertical line that intersects the graph more than once, then the graph is not a function. A function can only have one y-value for each x-value.

Consider the graph of y = ±√(x + 2). This graph is a sideways parabola with its vertex at (-2, 0). If you draw a vertical line at, say, x = 2, it will intersect the graph at two points: (2, 2) and (2, -2). This means that for the input x = 2, we have two outputs: y = 2 and y = -2. Because there is more than one output, according to the vertical line test, the inverse of g(x) is not a function.

Why the Inverse Is Not a Function: Multiple Outputs for One Input

The inverse of g(x) = x² - 2 is not a function because for each input of the inverse of g(x) there is more than one output.

In simpler terms, when we found the inverse, we ended up with y = ±√(x + 2). The ± sign is the culprit. It tells us that for a single x-value (except for x = -2), we get two different y-values. This violates the fundamental rule that a function can only have one unique output for each input. Thus, the inverse isn't a function.

Restricting the Domain to Make the Inverse a Function

Now, here’s a cool trick: We can make the inverse a function by restricting the domain of the original function. Remember how the original function, g(x) = x² - 2, gave us a parabola? The problem is that the parabola is symmetrical. What if we only considered half of the parabola?

Let’s say we restrict the domain of g(x) to x ≥ 0. This means we're only looking at the right side of the parabola. Now, if we find the inverse, we only consider the positive square root:

y = √(x + 2)

This is a function! Because we only took the positive square root, each x-value will only have one corresponding y-value. Graphically, we've taken the right half of the sideways parabola. This restricted inverse passes the vertical line test.

Alternatively, we could restrict the domain of g(x) to x ≤ 0, which means we're only looking at the left side of the parabola. In this case, when we find the inverse, we only consider the negative square root:

y = -√(x + 2)

This is also a function! We've taken the left half of the sideways parabola, and it also passes the vertical line test.

In Summary

So, to wrap it all up:

  • The inverse of g(x) = x² - 2 is initially y = ±√(x + 2).
  • The graph of the inverse is the reflection of the graph of g(x) across the line y = x.
  • The inverse of g(x) is not a function because for each input of the inverse of g(x) there is more than one output.
  • However, we can restrict the domain of the original function g(x) to x ≥ 0 or x ≤ 0 to make the inverse a function.

I hope this explanation helps you understand the intricacies of inverse functions and how to determine whether they are functions themselves. Keep exploring those mathematical concepts, you guys! You got this!