Inverse Function: Find G(x) Of F(x) = 11/(x-4)

$f(x)=\frac{11}{x-4}$

Finding the inverse of a function might seem daunting at first, but trust me, it's a pretty straightforward process once you get the hang of it. Basically, when we're looking for the inverse function, what we're really trying to do is undo what the original function does. So, if $f(x)$ takes an input $x$ and transforms it into some output, the inverse function, which we call $g(x)$ or $f^{-1}(x)$, takes that output and brings it right back to the original $x$. It's like a round trip! To nail this, let's dive into the steps with our given function, $f(x) = \frac{11}{x-4}$.

First things first, replace $f(x)$ with $y$. This makes the equation a bit easier to manipulate. So, we rewrite $f(x) = \frac{11}{x-4}$ as $y = \frac{11}{x-4}$. This is just a notational change, but it helps to clarify what we're doing. Now, the big trick to finding the inverse is to swap $x$ and $y$. This is the heart of the process because it reflects the idea that the inverse function undoes the original function. After swapping, we get $x = \frac{11}{y-4}$.

Next up, we need to solve for $y$. This means we want to isolate $y$ on one side of the equation. To do this, we first multiply both sides by $(y-4)$ to get rid of the fraction: $x(y-4) = 11$. Distribute the $x$ on the left side: $xy - 4x = 11$. Now, we want to isolate the term with $y$, so we add $4x$ to both sides: $xy = 11 + 4x$. Finally, we divide both sides by $x$ to solve for $y$: $y = \frac{11 + 4x}{x}$.

Now that we've solved for $y$, we can write the inverse function. We replace $y$ with $g(x)$, which represents the inverse function of $f(x)$. So, $g(x) = \frac{11 + 4x}{x}$. This is the inverse function we were looking for! You can also write it as $g(x) = \frac{11}{x} + 4$ if you prefer to separate the fraction. Remember, the inverse function $g(x)$ takes the output of $f(x)$ and returns the original input $x$.

To recap, finding the inverse function involves replacing $f(x)$ with $y$, swapping $x$ and $y$, solving for $y$, and then replacing $y$ with $g(x)$. For our function $f(x) = \frac{11}{x-4}$, the inverse function is $g(x) = \frac{11 + 4x}{x}$. This process might seem a bit abstract, but with practice, it becomes second nature. So keep at it, and you'll be finding inverse functions like a pro in no time!

Verification of the Inverse Function

To be absolutely sure that $g(x)$ is indeed the inverse of $f(x)$, we need to verify that $f(g(x)) = x$ and $g(f(x)) = x$. This confirms that applying one function and then the other brings us back to where we started. Let's start by computing $f(g(x))$.

We have $f(x) = \frac{11}{x-4}$ and $g(x) = \frac{11+4x}{x}$. So, $f(g(x)) = f(\frac{11+4x}{x}) = \frac{11}{\frac{11+4x}{x} - 4}$. To simplify the denominator, we need to find a common denominator, which is $x$. So, we rewrite $4$ as $\frac{4x}{x}$. Thus, the denominator becomes $\frac{11+4x}{x} - \frac{4x}{x} = \frac{11+4x-4x}{x} = \frac{11}{x}$. Therefore, $f(g(x)) = \frac{11}{\frac{11}{x}}$. To divide by a fraction, we multiply by its reciprocal, so $f(g(x)) = 11 \cdot \frac{x}{11} = x$.

Now, let's compute $g(f(x))$. We have $g(x) = \frac{11+4x}{x}$ and $f(x) = \frac{11}{x-4}$. So, $g(f(x)) = g(\frac{11}{x-4}) = \frac{11 + 4(\frac{11}{x-4})}{\frac{11}{x-4}}$. To simplify the numerator, we have $11 + 4(\frac{11}{x-4}) = 11 + \frac{44}{x-4}$. We need a common denominator to add these terms, so we rewrite $11$ as $\frac{11(x-4)}{x-4} = \frac{11x-44}{x-4}$. Thus, the numerator becomes $\frac{11x-44}{x-4} + \frac{44}{x-4} = \frac{11x-44+44}{x-4} = \frac{11x}{x-4}$. Therefore, $g(f(x)) = \frac{\frac{11x}{x-4}}{\frac{11}{x-4}}$. To divide by a fraction, we multiply by its reciprocal, so $g(f(x)) = \frac{11x}{x-4} \cdot \frac{x-4}{11} = \frac{11x(x-4)}{11(x-4)} = x$.

Since both $f(g(x)) = x$ and $g(f(x)) = x$, we have verified that $g(x) = \frac{11+4x}{x}$ is indeed the inverse function of $f(x) = \frac{11}{x-4}$. Verifying the inverse is a crucial step to ensure accuracy, especially in more complex problems. Always double-check!

Domain and Range Considerations

When dealing with functions and their inverses, it's essential to consider the domain and range. The domain of a function is the set of all possible input values, while the range is the set of all possible output values. For inverse functions, the domain of $f(x)$ is the range of $g(x)$, and the range of $f(x)$ is the domain of $g(x)$. This reciprocal relationship is a key aspect of inverse functions.

For our function $f(x) = \frac{11}{x-4}$, the domain is all real numbers except $x = 4$, because division by zero is undefined. So, the domain of $f(x)$ is $(-\infty, 4) \cup (4, \infty)$. The range of $f(x)$ is all real numbers except $y = 0$, because the fraction can never be zero. So, the range of $f(x)$ is $(-\infty, 0) \cup (0, \infty)$.

Now let's consider the inverse function $g(x) = \frac{11+4x}{x}$. The domain of $g(x)$ is all real numbers except $x = 0$, because again, division by zero is undefined. So, the domain of $g(x)$ is $(-\infty, 0) \cup (0, \infty)$. The range of $g(x)$ is all real numbers except $y = 4$. To see this, we can rewrite $g(x)$ as $g(x) = \frac{11}{x} + 4$. As $x$ approaches infinity, $\frac{11}{x}$ approaches 0, so $g(x)$ approaches 4, but never actually equals 4. So, the range of $g(x)$ is $(-\infty, 4) \cup (4, \infty)$.

Notice that the domain of $f(x)$ is the range of $g(x)$, and the range of $f(x)$ is the domain of $g(x)$, as expected. Understanding the domain and range helps us to fully grasp the behavior of functions and their inverses. Keep these concepts in mind when working with inverse functions!

Graphical Interpretation

Graphically, the inverse function $g(x)$ is the reflection of the original function $f(x)$ across the line $y = x$. This means that if you were to draw the graph of $f(x)$ and then flip it over the line $y = x$, you would get the graph of $g(x)$. This graphical interpretation provides another way to visualize the relationship between a function and its inverse.

For our function $f(x) = \frac{11}{x-4}$, the graph is a hyperbola with a vertical asymptote at $x = 4$ and a horizontal asymptote at $y = 0$. The graph of the inverse function $g(x) = \frac{11+4x}{x}$ is also a hyperbola, but with a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 4$. If you were to plot these two graphs on the same coordinate plane, you would see that they are reflections of each other across the line $y = x$.

Understanding the graphical relationship between a function and its inverse can be a powerful tool for visualizing and verifying your results. It's always a good idea to sketch the graphs of the functions to get a better sense of their behavior. Visualizing functions can make abstract concepts much more concrete!

In conclusion, finding the inverse function involves swapping $x$ and $y$, solving for $y$, and verifying that $f(g(x)) = x$ and $g(f(x)) = x$. We also need to consider the domain and range of the functions and understand the graphical interpretation. For the function $f(x) = \frac{11}{x-4}$, the inverse function is $g(x) = \frac{11+4x}{x}$. Keep practicing, and you'll master the art of finding inverse functions in no time!