Are F(x) And G(x) Inverse Functions?

Let's dive into the fascinating world of functions and their inverses! You've likely encountered functions in your math journey, those mathematical machines that take an input and give you an output. But have you ever wondered if there's a way to "undo" what a function does? That's where the concept of inverse functions comes in. In this article, we'll explore how to determine whether two functions, specifically $f(x) = 2x - 10$ and $g(x) = \frac{x}{2} + 5$, are inverses of each other using the fundamental definition of inverses. This definition is our guiding star, our mathematical compass, that helps us navigate the relationship between a function and its potential inverse. Without understanding this core principle, identifying inverse functions would be like trying to solve a puzzle without the picture on the box – confusing and ultimately unachievable. We'll break down the definition, apply it step-by-step to our given functions, and see if they indeed fit the criteria of being perfect opposites. Get ready to demystify inverse functions and gain a solid understanding of this crucial mathematical concept.

Understanding the Definition of Inverse Functions

The definition of inverse functions is quite elegant and serves as the cornerstone for our investigation. For two functions, $f$ and $g$, to be considered inverses of each other, they must satisfy a specific condition: when you compose one function with the other, the result should be the identity function. The identity function is super simple; it just returns the input value unchanged. Mathematically, this means two things must be true:

1. $f(g(x)) = x$ for all $x$ in the domain of $g$.
2. $g(f(x)) = x$ for all $x$ in the domain of $f$.

Think of it this way: if $f$ is a function that adds 5 to a number, its inverse $g$ should be the function that subtracts 5 from a number. If you take a number, add 5 to it (using $f$), and then subtract 5 from the result (using $g$), you should get your original number back. This is the essence of the composition that leads to the identity function, $x$. This definition is crucial because it provides a concrete, testable method for verifying the inverse relationship. It's not just a vague idea; it's a set of mathematical operations that must yield a specific outcome. We're not guessing; we're proving. The domain conditions are also important to remember, as they ensure that the composition is valid across the relevant input values. If either of these composition conditions fails, then $f$ and $g$ are not inverses, no matter how similar they might appear. This rigorous definition ensures that we can confidently identify true inverse pairs and distinguish them from functions that merely resemble inverses.

Applying the Definition to $f(x)$ and $g(x)$

Now, let's roll up our sleeves and apply this definition of inverse functions to our specific pair: $f(x) = 2x - 10$ and $g(x) = \frac{x}{2} + 5$. We need to perform two composition tests. The first test is to compute $f(g(x))$. Remember, when we see $f(g(x))$, it means we take the entire function $g(x)$ and substitute it wherever we see an '$x$' in the function $f(x)$. So, we start with $f(x) = 2x - 10$. Now, we replace that '$x$' with $g(x)$, which is $\frac{x}{2} + 5$. This gives us:

$f(g(x)) = 2 \left( \frac{x}{2} + 5 \right) - 10$

See how we substituted the expression for $g(x)$ into $f(x)$? The next step is to simplify this expression. We'll distribute the 2 to both terms inside the parentheses:

$f(g(x)) = 2 \cdot \frac{x}{2} + 2 \cdot 5 - 10$

$f(g(x)) = x + 10 - 10$

And simplifying further, we get:

$f(g(x)) = x$

Fantastic! The first condition is met. When we composed $f$ with $g$, we indeed got $x$. This is a very promising sign that $f$ and $g$ might be inverses. However, we're not done yet. The definition requires both conditions to be true. So, we must proceed to the second test.

The Second Composition Test: $g(f(x))$

Our second test involves computing $g(f(x))$. This is similar to the first test, but this time, we take the function $f(x)$ and substitute it into the function $g(x)$. Our function $g(x)$ is defined as $g(x) = \frac{x}{2} + 5$. We will replace the '$x$' in $g(x)$ with the entire expression for $f(x)$, which is $2x - 10$. So, we get:

$g(f(x)) = \frac{(2x - 10)}{2} + 5$

Again, the key is that we're substituting the whole of $f(x)$ into $g(x)$. Now, let's simplify this expression. We can divide each term in the numerator by 2:

$g(f(x)) = \frac{2x}{2} - \frac{10}{2} + 5$

Performing the division, we get:

$g(f(x)) = x - 5 + 5$

And simplifying further, we arrive at:

$g(f(x)) = x$

Incredible! The second condition is also met. Just like in the first test, when we composed $g$ with $f$, we ended up with $x$. This means that both parts of the definition of inverse functions have been satisfied.

Conclusion: Are $f(x)$ and $g(x)$ Inverses?

Since both $f(g(x)) = x$ and $g(f(x)) = x$ are true for the functions $f(x) = 2x - 10$ and $g(x) = \frac{x}{2} + 5$, we can definitively conclude that $f$ and $g$ are indeed inverse functions. They perfectly "undo" each other's operations, as demonstrated by the identity function result in both composition tests. This rigorous application of the definition of inverses confirms their reciprocal relationship. It's satisfying to see how algebraic manipulation can confirm such a fundamental concept in mathematics. If either of those compositions had resulted in anything other than $x$, we would have had to conclude that they are not inverses. But in this case, they are a perfect pair!

Understanding inverse functions is a fundamental skill in algebra and calculus, enabling us to solve equations, analyze transformations, and much more. If you'd like to explore more about functions and their properties, the Khan Academy website offers excellent resources and practice problems on this topic. For a deeper dive into the theoretical aspects of functions and their inverses, the Mathematics LibreTexts project provides comprehensive explanations and examples.

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