Finding The Inverse: H⁻¹(x) Explained With Examples

Hey everyone! Today, we're diving into the fascinating world of inverse functions, specifically how to find the inverse of a function when it's presented as a set of ordered pairs. We'll be working through an example step-by-step, so you can easily grasp the concept and apply it to similar problems. So, let's get started and figure out how to tackle this! We'll break it down so it's super easy to follow.

The Question: Finding the Inverse Function

The problem we're tackling today presents us with a function, helpfully named $h(x)$, which is defined as a set of ordered pairs:

$h(x) = \{(3, -5), (5, -7), (6, -9), (10, -12), (12, -16)\}$

The burning question is: Which of the following options correctly represents the inverse function, denoted as $h^{-1}(x)$?

Here are the options we need to consider:

• A. $\{(3, 5), (5, 7), (6, 9), (10, 12), (12, 16)\}$
• B. $\{(-5, 3), (-7, 5), (-9, 6), (-12, 10), (-16, 12)\}$
• C. $\{(3, -5), (5, -7), (6, -9), (10, -12), (12, -16)\}$
• D. Discussion category: mathematics

Before we jump into the solution, let's take a moment to really understand what an inverse function is all about. This foundational knowledge is key to solving this problem and others like it.

What is an Inverse Function? A Deep Dive

So, what exactly is an inverse function? Inverse functions are like the reverse button on a mathematical operation. If a function $f(x)$ takes an input $x$ and produces an output $y$, its inverse function, denoted as $f^{-1}(x)$, takes that output $y$ and magically returns the original input $x$. Think of it as undoing what the original function did.

To put it simply, if $f(a) = b$, then $f^{-1}(b) = a$. This is the core principle behind inverse functions. The input and output swap places! This swapping of inputs and outputs is what we'll leverage to find the inverse of our given function, $h(x)$.

Let's break this down further with an analogy. Imagine a machine that takes fruit as input and outputs juice. That's our function, $f(x)$. The inverse function, $f^{-1}(x)$, would be a machine that takes juice as input and somehow reconstructs the original fruit! Okay, maybe not literally, but you get the idea. It reverses the process.

Graphically, this input-output swap has a beautiful consequence. The graph of a function and its inverse are reflections of each other across the line $y = x$. This means if you were to fold the graph along the line $y = x$, the function and its inverse would perfectly overlap. This visual representation can be incredibly helpful in understanding the relationship between a function and its inverse.

Now, back to ordered pairs. When a function is presented as a set of ordered pairs, like our $h(x)$, finding the inverse becomes remarkably straightforward. Since the inverse function swaps inputs and outputs, all we need to do is swap the $x$ and $y$ values in each ordered pair. That's it! Seems simple, right? Let's put this knowledge into action.

Finding h⁻¹(x): The Step-by-Step Solution

Alright, guys, let's roll up our sleeves and actually find $h^{-1}(x)$. Remember, our function $h(x)$ is given as:

$h(x) = \{(3, -5), (5, -7), (6, -9), (10, -12), (12, -16)\}$

The key to finding the inverse is to swap the $x$ and $y$ coordinates in each ordered pair. So, let's go through each pair one by one:

• (3, -5): Swapping the coordinates gives us (-5, 3).
• (5, -7): Swapping the coordinates gives us (-7, 5).
• (6, -9): Swapping the coordinates gives us (-9, 6).
• (10, -12): Swapping the coordinates gives us (-12, 10).
• (12, -16): Swapping the coordinates gives us (-16, 12).

Now, let's collect these new ordered pairs to form our inverse function, $h^{-1}(x)$:

$h^{-1}(x) = \{(-5, 3), (-7, 5), (-9, 6), (-12, 10), (-16, 12)\}$

Drumroll please! We've found it! Now, let's compare this result with the options provided in the question.

Matching the Solution with the Options

Looking back at the options, we can see that our calculated inverse function, $h^{-1}(x) = \{(-5, 3), (-7, 5), (-9, 6), (-12, 10), (-16, 12)\}$, perfectly matches option B.

• A. $\{(3, 5), (5, 7), (6, 9), (10, 12), (12, 16)\}$ - Incorrect. This option simply changes the signs of the $y$-coordinates but doesn't swap them with the $x$-coordinates.
• B. $\{(-5, 3), (-7, 5), (-9, 6), (-12, 10), (-16, 12)\}$ - Correct! This option accurately represents the inverse function, $h^{-1}(x)$.
• C. $\{(3, -5), (5, -7), (6, -9), (10, -12), (12, -16)\}$ - Incorrect. This option is the same as the original function, $h(x)$, and doesn't represent the inverse.
• D. Discussion category: mathematics - This isn't an answer choice; it's just a category.

Therefore, the correct answer is B. We've successfully found the inverse function by swapping the $x$ and $y$ coordinates in each ordered pair. Awesome job, everyone!

Key Takeaways and Common Mistakes to Avoid

Before we wrap up, let's quickly recap the key takeaways from this problem and discuss some common mistakes to avoid when dealing with inverse functions.

• The Definition of an Inverse Function: Remember that the inverse function, $f^{-1}(x)$, essentially undoes the original function, $f(x)$. If $f(a) = b$, then $f^{-1}(b) = a$.
• Swapping Coordinates for Ordered Pairs: When a function is given as a set of ordered pairs, finding the inverse is as simple as swapping the $x$ and $y$ coordinates in each pair.
• Graphical Representation: Visualize inverse functions as reflections of each other across the line $y = x$. This can help you understand their relationship intuitively.

Now, let's talk about common mistakes. One frequent error is simply changing the signs of the coordinates instead of swapping them. Option A in our problem is a perfect example of this. It's crucial to remember that finding the inverse involves swapping the positions of the $x$ and $y$ values, not just altering their signs.

Another mistake is confusing the inverse function with the reciprocal. The notation $f^{-1}(x)$ might trick some into thinking it means $\frac{1}{f(x)}$, but these are entirely different concepts. The inverse function undoes the operation, while the reciprocal is simply 1 divided by the function's value.

Finally, always double-check your work! After finding the inverse, you can verify your answer by applying the function to the inverse function (or vice versa). The result should be $x$. For example, if you have $f(x)$ and $f^{-1}(x)$, then $f(f^{-1}(x))$ should equal $x$.

By keeping these takeaways and potential pitfalls in mind, you'll be well-equipped to tackle inverse function problems with confidence!

Practice Makes Perfect: Further Exploration

Okay, guys, we've covered a lot today, but the best way to solidify your understanding is through practice. Try working through similar problems with different sets of ordered pairs. You can even create your own functions and find their inverses. The more you practice, the more comfortable you'll become with the concept.

Consider exploring these avenues for further practice:

• Textbook Exercises: Check your textbook for practice problems on inverse functions. Most textbooks have a section dedicated to this topic.
• Online Resources: Websites like Khan Academy and Mathway offer a wealth of practice problems and explanations.
• Worksheets: Search online for printable worksheets on inverse functions. This can be a great way to test your knowledge.

Don't be afraid to challenge yourself with more complex functions, such as those involving equations. The fundamental principle of swapping inputs and outputs still applies, but you'll need to use algebraic techniques to solve for the inverse function.

Remember, learning mathematics is a journey, not a destination. Keep exploring, keep practicing, and most importantly, keep asking questions! You've got this!

By understanding the core concept of inverse functions and practicing consistently, you'll be able to confidently solve problems like this one and excel in your math studies. Keep up the great work!