How To Find The Inverse Of Functions F(x) = 4x - 12 And H(x) = (2x - 4)/3
Hey guys! Ever wondered how to undo a function? That's where inverse functions come in! Think of it like reversing a recipe – you're starting with the final product and trying to figure out the original ingredients and steps. In math terms, if a function takes an input x and gives you an output y, the inverse function takes that y and spits back the original x. Today, we're going to dive deep into finding the inverses of two specific functions: f(x) = 4x - 12 and h(x) = (2x - 4)/3. Buckle up, because we're about to embark on a mathematical adventure!
Understanding Inverse Functions
Before we jump into solving, let's make sure we're all on the same page about what inverse functions actually are. An inverse function essentially "undoes" what the original function does. If we apply a function and then its inverse (or vice versa), we should end up back where we started. Mathematically, this means that if f(x) = y, then the inverse function, denoted as f⁻¹(y), should equal x. This is the core concept we'll be using to find our inverses.
To further illustrate this, imagine a function as a machine that takes an input, processes it, and gives an output. The inverse function is like a machine that takes the output and reverses the process to give us the original input. For example, if our function is f(x) = x + 5, it adds 5 to any input. The inverse function, therefore, would subtract 5 from any input, which can be written as f⁻¹(x) = x - 5. If we put a number, say 3, into the original function, we get f(3) = 3 + 5 = 8. Now, if we put 8 into the inverse function, we get f⁻¹(8) = 8 - 5 = 3, which is our original input! This confirms that the inverse function correctly reverses the operation of the original function.
There are a couple of important things to keep in mind about inverse functions. First, not every function has an inverse. For a function to have an inverse, it must be one-to-one. A one-to-one function means that each input corresponds to a unique output, and each output corresponds to a unique input. Graphically, this can be checked using the horizontal line test: if any horizontal line intersects the graph of the function more than once, the function is not one-to-one and does not have an inverse. Second, the domain of the inverse function is the range of the original function, and the range of the inverse function is the domain of the original function. This makes sense because the inverse function is essentially swapping the inputs and outputs.
Finding the inverse of a function involves a few key steps. We'll be using these steps to solve for the inverses of f(x) and h(x). Let's outline these steps before we dive in:
- Replace f(x) or h(x) with y: This makes the equation easier to manipulate.
- Swap x and y: This is the crucial step in finding the inverse, as we're reversing the roles of input and output.
- Solve for y: Isolate y on one side of the equation. This will give us the equation for the inverse function.
- Replace y with f⁻¹(x) or h⁻¹(x): This is just notational convention to show that we've found the inverse function.
With these concepts and steps in mind, we're now ready to tackle the given functions and find their inverses. Let's start with f(x) = 4x - 12.
Finding the Inverse of f(x) = 4x - 12
Okay, let's get our hands dirty and find the inverse of our first function, f(x) = 4x - 12. Remember the steps we just outlined? We're going to follow them meticulously to make sure we get the correct inverse.
Step 1: Replace f(x) with y
This is a straightforward substitution. We simply replace f(x) with y in our equation. So, f(x) = 4x - 12 becomes:
y = 4x - 12
This step makes the equation look a bit more familiar and easier to work with. It's purely a notational change, but it helps to clarify the roles of x and y as input and output.
Step 2: Swap x and y
This is the key step in finding the inverse. We're essentially reversing the roles of input and output. Wherever we see an x, we replace it with a y, and wherever we see a y, we replace it with an x. So, our equation y = 4x - 12 becomes:
x = 4y - 12
This step is crucial because it reflects the fundamental concept of an inverse function: it undoes the original function by swapping the input and output. We've now set up the equation to solve for y, which will give us the inverse function.
Step 3: Solve for y
Now comes the algebraic manipulation! Our goal is to isolate y on one side of the equation. To do this, we'll perform a series of operations, making sure to maintain the equality of the equation.
We have the equation x = 4y - 12. First, we want to get rid of the -12. We can do this by adding 12 to both sides of the equation:
x + 12 = 4y - 12 + 12
This simplifies to:
x + 12 = 4y
Next, we want to isolate y, which is currently being multiplied by 4. To undo this multiplication, we'll divide both sides of the equation by 4:
(x + 12) / 4 = 4y / 4
This simplifies to:
(x + 12) / 4 = y
We can also write this as:
y = (x + 12) / 4
We've successfully isolated y! This equation gives us the inverse function in terms of x.
Step 4: Replace y with f⁻¹(x)
This is the final step, and it's simply a notational change. We replace y with f⁻¹(x) to indicate that we've found the inverse function. So, y = (x + 12) / 4 becomes:
f⁻¹(x) = (x + 12) / 4
And there you have it! We've found the inverse of f(x) = 4x - 12. The inverse function is f⁻¹(x) = (x + 12) / 4. We can also simplify this further by distributing the division: f⁻¹(x) = x/4 + 3.
To verify that this is indeed the inverse function, we can compose the original function and its inverse. This means plugging the inverse function into the original function and vice versa. If we get x as the result in both cases, we've found the correct inverse. Let's try it out:
f(f⁻¹(x)) = f((x + 12) / 4) = 4((x + 12) / 4) - 12 = (x + 12) - 12 = x
f⁻¹(f(x)) = f⁻¹(4x - 12) = ((4x - 12) + 12) / 4 = (4x) / 4 = x
Since both compositions result in x, we've confirmed that f⁻¹(x) = (x + 12) / 4 is the correct inverse function. Awesome work! Now, let's move on to the next function, h(x) = (2x - 4) / 3, and find its inverse.
Finding the Inverse of h(x) = (2x - 4) / 3
Alright, let's tackle our second function, h(x) = (2x - 4) / 3. We'll follow the same four steps we used before, but this time with a slightly different function. Remember, the key is to stay organized and follow each step carefully.
Step 1: Replace h(x) with y
Just like before, we start by replacing h(x) with y. This gives us:
y = (2x - 4) / 3
This step simplifies the notation and makes the equation easier to work with.
Step 2: Swap x and y
Now, we swap x and y. This is the heart of finding the inverse, as it reverses the roles of input and output. Our equation y = (2x - 4) / 3 becomes:
x = (2y - 4) / 3
We've now set up the equation to solve for y, which will give us the inverse function.
Step 3: Solve for y
This step involves a bit more algebraic manipulation than the previous example, but we can handle it! Our goal is to isolate y on one side of the equation. We have:
x = (2y - 4) / 3
First, let's get rid of the fraction by multiplying both sides of the equation by 3:
3 * x = 3 * ((2y - 4) / 3)
This simplifies to:
3x = 2y - 4
Next, we want to isolate the term with y, so we'll add 4 to both sides of the equation:
3x + 4 = 2y - 4 + 4
This simplifies to:
3x + 4 = 2y
Finally, we want to isolate y, which is currently being multiplied by 2. To undo this multiplication, we'll divide both sides of the equation by 2:
(3x + 4) / 2 = 2y / 2
This simplifies to:
(3x + 4) / 2 = y
We can also write this as:
y = (3x + 4) / 2
We've successfully isolated y! This equation gives us the inverse function in terms of x.
Step 4: Replace y with h⁻¹(x)
Our final step is to replace y with h⁻¹(x) to denote that we've found the inverse function. So, y = (3x + 4) / 2 becomes:
h⁻¹(x) = (3x + 4) / 2
And there we have it! The inverse of h(x) = (2x - 4) / 3 is h⁻¹(x) = (3x + 4) / 2. We can also rewrite this as h⁻¹(x) = (3/2)x + 2.
To be absolutely sure we've found the correct inverse, let's compose the original function and its inverse, just like we did before:
h(h⁻¹(x)) = h((3x + 4) / 2) = (2((3x + 4) / 2) - 4) / 3 = (3x + 4 - 4) / 3 = (3x) / 3 = x
h⁻¹(h(x)) = h⁻¹((2x - 4) / 3) = (3((2x - 4) / 3) + 4) / 2 = (2x - 4 + 4) / 2 = (2x) / 2 = x
Again, both compositions result in x, confirming that h⁻¹(x) = (3x + 4) / 2 is indeed the correct inverse function. Fantastic!
Conclusion
So, guys, we've successfully navigated the world of inverse functions and found the inverses of f(x) = 4x - 12 and h(x) = (2x - 4) / 3. We learned that the inverse of f(x) = 4x - 12 is f⁻¹(x) = (x + 12) / 4 (or f⁻¹(x) = x/4 + 3), and the inverse of h(x) = (2x - 4) / 3 is h⁻¹(x) = (3x + 4) / 2 (or h⁻¹(x) = (3/2)x + 2).
The key takeaway here is the systematic approach to finding inverse functions: replace f(x) or h(x) with y, swap x and y, solve for y, and replace y with f⁻¹(x) or h⁻¹(x). By following these steps carefully, you can confidently find the inverse of many different functions. Remember to always double-check your work by composing the original function and its inverse – if you get x, you know you've done it right!
Understanding inverse functions is a fundamental concept in mathematics, and it opens the door to exploring more advanced topics. So, keep practicing, keep exploring, and keep having fun with math! You've got this!
