Inverse Function Of F(x) = (x+2)/7 Step-by-Step Solution
Hey guys! Today, we're diving into the fascinating world of inverse functions. Specifically, we're going to tackle the problem of finding the inverse of the function $f(x) = \frac{x+2}{7}$. Don't worry if this looks a bit intimidating at first; we'll break it down step-by-step, making it super easy to understand. So, grab your thinking caps, and let's get started!
Understanding Inverse Functions
Before we jump into solving the problem, let's quickly recap what an inverse function actually is. Think of a function like a machine: you feed it an input (x), and it spits out an output (f(x)). An inverse function is like a machine that does the exact opposite. If you feed it the output (f(x)), it spits out the original input (x). In mathematical terms, if we have a function $f(x)$, its inverse is denoted as $f^-1}(x)$. The key relationship between a function and its inverse is this(f(x)) = x$ and $f(f^{-1}(x)) = x$. This means that if you apply a function and then its inverse (or vice versa), you end up back where you started.
Now, let’s talk about how we actually find these inverse functions. The general process involves a few key steps, which we'll illustrate with our example function. First, we replace $f(x)$ with $y$. This makes the equation easier to manipulate. Second, we swap the positions of $x$ and $y$. This is the crucial step that reflects the inverse relationship. Third, we solve the equation for $y$. This isolates the inverse function. Finally, we replace $y$ with $f^{-1}(x)$ to express the inverse function in standard notation. This process might seem a bit abstract right now, but it will become crystal clear as we work through our example.
Remember, not every function has an inverse. For a function to have an inverse, it must be one-to-one. This means that each input (x) produces a unique output (f(x)), and each output (f(x)) corresponds to a unique input (x). Graphically, a one-to-one function passes the horizontal line test: no horizontal line intersects the graph more than once. Functions that aren't one-to-one can sometimes be made one-to-one by restricting their domain, but that's a topic for another day. For our function, $f(x) = \frac{x+2}{7}$, it is indeed one-to-one, so we can confidently proceed with finding its inverse.
Finding the Inverse of f(x) = (x+2)/7
Okay, let's get down to business and find the inverse function of $f(x) = \frac{x+2}{7}$. We'll follow the steps we outlined earlier to make sure we get it right. This is where the fun really begins, so pay close attention, guys! We're about to turn this mathematical puzzle into a piece of cake.
Step 1: Replace f(x) with y
This is a simple substitution to make the equation easier to work with. So, we rewrite our function as: $y = \frac{x+2}{7}$. See? Nothing too scary here. We're just setting the stage for the real work.
Step 2: Swap x and y
This is the heart of finding the inverse. We're reflecting the function across the line $y = x$, which is the graphical representation of finding an inverse. By swapping $x$ and $y$, we get: $x = \frac{y+2}{7}$. This new equation represents the inverse relationship, but we need to solve it for $y$ to express it in the standard form of a function.
Step 3: Solve for y
Now comes the algebra! 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. First, we multiply both sides of the equation by 7 to get rid of the fraction: $7x = y + 2$. Next, we subtract 2 from both sides to isolate $y$: $7x - 2 = y$. We've successfully solved for $y$! This tells us what the inverse function looks like, but we need to express it using the proper notation.
Step 4: Replace y with f⁻¹(x)
This is the final step, where we express our answer using the standard notation for inverse functions. We replace $y$ with $f^-1}(x)$, giving us(x) = 7x - 2$. And there you have it! We've found the inverse function of $f(x) = \frac{x+2}{7}$. It's as simple as that, guys!
Analyzing the Answer Choices
Now that we've found the inverse function, let's take a look at the answer choices provided and see which one matches our result. This is a crucial step to ensure we've done everything correctly and haven't made any silly mistakes along the way. Remember, double-checking your work is always a good idea, especially in math!
The answer choices were:
A. $s(x) = 2x + 7$ B. $r(x) = \frac{7}{x+2}$ C. $q(x) = \frac{-x+2}{7}$ D. $p(x) = 7x - 2$
Comparing our result, $f^{-1}(x) = 7x - 2$, with the answer choices, we can clearly see that option D, $p(x) = 7x - 2$, is the correct answer. Woohoo! We nailed it! This confirms that our step-by-step process for finding the inverse function was accurate and effective.
The other options are incorrect. Option A has the terms in the wrong order and the wrong coefficients. Option B represents the reciprocal of a related function, not the inverse. Option C has the wrong sign for the x term. It's important to understand why these options are incorrect to solidify your understanding of inverse functions. Remember, the inverse function undoes the original function, and only option D achieves this.
Conclusion: Mastering Inverse Functions
Great job, everyone! We've successfully found the inverse function of $f(x) = \frac{x+2}{7}$ and identified the correct answer choice. You've now got a solid grasp on the process of finding inverse functions, which is a valuable skill in mathematics.
Remember, the key steps are: replace $f(x)$ with $y$, swap $x$ and $y$, solve for $y$, and replace $y$ with $f^{-1}(x)$. By following these steps carefully, you can tackle any inverse function problem that comes your way. And don't forget to double-check your answer by verifying that $f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$.
So, keep practicing, keep exploring, and keep having fun with math! You've got this, guys! Understanding inverse functions opens up a whole new world of mathematical possibilities, and you're well on your way to mastering them. Keep up the awesome work, and I'll see you in the next math adventure!
