Inverse Function: Find F⁻¹(x) For F(x) = X² - 16
Hey everyone! Today, we're diving into a super important concept in mathematics: finding the inverse of a function. Specifically, we're going to tackle a problem where we need to find the inverse of the function f(x) = x² - 16, but with a little twist – the domain of f(x) is restricted to x ≥ 0. Don't worry if that sounds complicated; we'll break it down step by step. Understanding inverses is crucial because it allows us to "undo" what a function does, and they pop up in various areas of math and science.
Understanding Inverse Functions
Before we jump into the problem, let's make sure we're all on the same page about what an inverse function actually is. Think of a function like a machine: you put something in (the input), and it spits something else out (the output). An inverse function is like a machine that does the reverse – it takes the output of the original function and gives you back the original input. Mathematically, if f(a) = b, then the inverse function, denoted as f⁻¹(x), would satisfy f⁻¹(b) = a. This "undoing" relationship is the heart of inverse functions.
However, there's a catch! Not all functions have inverses. For a function to have an inverse, it must be one-to-one. A function is one-to-one if each output corresponds to only one input. Graphically, this means the function passes the horizontal line test: any horizontal line drawn on the graph intersects the function at most once. If a function isn't one-to-one, we might need to restrict its domain (the set of possible inputs) to make it one-to-one and thus have an inverse. This is exactly what's happening in our problem with f(x) = x² - 16.
Why the Domain Restriction Matters
You might be wondering why the domain restriction x ≥ 0 is so important. Well, let's think about the graph of f(x) = x² - 16. It's a parabola that opens upwards, with its vertex (the lowest point) at (0, -16). If we consider the entire parabola, it's clear that it doesn't pass the horizontal line test. For example, the horizontal line y = 0 intersects the parabola at two points: x = 4 and x = -4. This means that the output 0 corresponds to two different inputs, so the function isn't one-to-one over its entire domain. That's why we restrict the domain to x ≥ 0. By only considering the right half of the parabola, we ensure that each output corresponds to only one input, making the function one-to-one and allowing us to find an inverse. This restriction is crucial for the inverse to be a well-defined function itself.
Solving the Problem: Finding the Inverse
Okay, now that we understand the concept of inverse functions and the importance of domain restrictions, let's get down to business and find the inverse of f(x) = x² - 16 with the domain x ≥ 0. Here's the general process for finding the inverse of a function:
- Replace f(x) with y: This is just a notational change to make the algebra a bit easier. So, we rewrite f(x) = x² - 16 as y = x² - 16.
- Swap x and y: This is the key step in finding the inverse. We're essentially reversing the roles of input and output. Swapping x and y gives us x = y² - 16.
- Solve for y: Now, we need to isolate y on one side of the equation. This will give us the equation for the inverse function. Let's add 16 to both sides: x + 16 = y². Next, we take the square root of both sides: √(x + 16) = y or -√(x + 16) = y. But here's where our domain restriction comes into play! Since the original function had a domain of x ≥ 0, the range of the inverse function (the possible outputs) must also be non-negative. Therefore, we only consider the positive square root: y = √(x + 16).
- Replace y with f⁻¹(x): This is just the final notational step to express our answer in standard inverse function notation. So, we have f⁻¹(x) = √(x + 16).
Therefore, the inverse of f(x) = x² - 16 with the domain x ≥ 0 is f⁻¹(x) = √(x + 16). So, the correct answer is A. f⁻¹(x) = √(x + 16).
A Closer Look at the Options
Let's quickly examine why the other options are incorrect. This can help solidify our understanding of inverse functions.
- B. f⁻¹(x) = √x + 4: This option is close, but it's missing a crucial detail. We need to add 16 before taking the square root, not after. This option would correspond to the inverse of a different function.
- C. f⁻¹(x) = √(x - 16): This option has the correct square root operation, but it subtracts 16 instead of adding it. This would be the inverse of f(x) = x² + 16, not f(x) = x² - 16.
By understanding why these options are incorrect, we gain a deeper understanding of the process of finding inverse functions and the importance of each step.
Verifying the Inverse Function
To be absolutely sure we've found the correct inverse function, we can verify it. Remember, if f⁻¹(x) is truly the inverse of f(x), then f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. Let's check both of these:
- f(f⁻¹(x)): We substitute f⁻¹(x) = √(x + 16) into f(x) = x² - 16: f(√(x + 16)) = (√(x + 16))² - 16 = (x + 16) - 16 = x. So, the first condition is satisfied.
- f⁻¹(f(x)): We substitute f(x) = x² - 16 into f⁻¹(x) = √(x + 16): f⁻¹(x² - 16) = √((x² - 16) + 16) = √(x²) = |x|. Since we have the domain restriction x ≥ 0, |x| = x. So, the second condition is also satisfied.
Both conditions hold true, confirming that f⁻¹(x) = √(x + 16) is indeed the inverse function.
Key Takeaways
Let's recap the key concepts we've covered in this article:
- Inverse functions