Verifying Inverse Functions: F(x) = 5x^2 - 2

In mathematics, inverse functions are functions that "undo" each other. If you apply a function and then its inverse, you should end up back where you started. This article dives deep into how to verify if two given functions, specifically $f(x) = 5x^2 - 2$ and $f^{-1}(x) = \pm\sqrt{\frac{x+2}{5}}$, are indeed inverse functions of each other. We will explore the concept of inverse functions, the methods to verify them, and work through the given example step by step. Understanding inverse functions is crucial in various areas of mathematics, including calculus, algebra, and analysis. Let’s embark on this mathematical journey to clarify this concept thoroughly. Understanding how to verify inverse functions is a fundamental concept in mathematics, especially in algebra and calculus. Two functions, $f(x)$ and $g(x)$, are considered inverses if $f(g(x)) = x$ and $g(f(x)) = x$ for all $x$ in their respective domains. This means that if you apply one function and then apply its inverse, you will return to the original input value. In simpler terms, the inverse function "undoes" what the original function does. For example, if $f(x)$ multiplies $x$ by 2, then its inverse, $f^{-1}(x)$, would divide $x$ by 2. The process of verifying inverse functions involves several key steps, including understanding the definition of inverse functions, computing the composition of the functions in both orders ($f(g(x))$ and $g(f(x))$), and ensuring that the result of both compositions is equal to $x$. Attention to detail is crucial in this process, as incorrect algebraic manipulation can lead to incorrect conclusions.

Understanding Inverse Functions

To determine if two functions are inverses, a crucial concept to grasp is the definition of inverse functions. Two functions, let's say f(x) and g(x), are deemed inverses of each other if applying one function and then the other results in the original input. Mathematically, this is expressed as:

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

This means that if you start with a value x, apply the function g to it, and then apply the function f to the result, you should end up with the original value x. The same holds true if you apply f first and then g. This “undoing” property is the hallmark of inverse functions. For instance, consider the functions $f(x) = 2x$ and $g(x) = \frac{x}{2}$. If you take any number, say 5, and apply $f(x)$, you get $f(5) = 10$. Now, if you apply $g(x)$ to this result, you get $g(10) = \frac{10}{2} = 5$, which is the original number. Similarly, if you apply $g(x)$ first, $g(5) = \frac{5}{2}$, and then apply $f(x)$, you get $f(\frac{5}{2}) = 2 \cdot \frac{5}{2} = 5$. This simple example illustrates the core principle of inverse functions. In contrast, if we had functions like $f(x) = x + 3$ and $g(x) = x - 1$, they would not be inverses because applying one after the other does not return the original input. For example, if we start with $x = 4$, $f(4) = 4 + 3 = 7$, and $g(7) = 7 - 1 = 6$, which is not the original 4. Understanding this fundamental concept is essential for verifying whether the given functions, $f(x) = 5x^2 - 2$ and $f^{-1}(x) = \pm\sqrt{\frac{x+2}{5}}$, are indeed inverses of each other. The verification process involves substituting one function into the other and checking if the result simplifies to x. This step-by-step approach ensures a clear and accurate determination of the inverse relationship.

Verifying $f(f^{-1}(x)) = x$

The first step in verifying if $f(x) = 5x^2 - 2$ and $f^{-1}(x) = \pm\sqrt{\frac{x+2}{5}}$ are inverse functions is to compute $f(f^{-1}(x))$. This involves substituting the inverse function $f^{-1}(x)$ into the original function $f(x)$ and simplifying the expression to see if it equals x. This process demonstrates whether applying the inverse function after the original function will return the initial input, which is a fundamental criterion for inverse functions. To begin, let’s write down the functions:

• $f(x) = 5x^2 - 2$
• $f^{-1}(x) = \pm\sqrt{\frac{x+2}{5}}$

Now, we substitute $f^{-1}(x)$ into $f(x)$:

$f(f^{-1}(x)) = 5(\pm\sqrt{\frac{x+2}{5}})^2 - 2$

When we square the square root, we get:

$f(f^{-1}(x)) = 5(\frac{x+2}{5}) - 2$

Notice that the $\pm$ sign disappears because squaring a positive or negative value results in a positive value. Next, we simplify the expression:

$f(f^{-1}(x)) = (x + 2) - 2$

$f(f^{-1}(x)) = x$

This result shows that $f(f^{-1}(x))$ indeed simplifies to x. However, to definitively confirm that the functions are inverses, we must also verify that $f^{-1}(f(x)) = x$. This is because the definition of inverse functions requires that both compositions result in x. Therefore, while this step is promising, it is only half of the verification process. The fact that $f(f^{-1}(x)) = x$ indicates that the inverse function correctly undoes the operation of the original function when applied in this order. This is a critical aspect of inverse functions, ensuring that the composite function returns the original input. The next step, verifying $f^{-1}(f(x)) = x$, is equally important to ensure the functions behave as inverses in both directions. This comprehensive approach ensures that no algebraic nuances are overlooked and that the inverse relationship is thoroughly confirmed.

Verifying $f^{-1}(f(x)) = x$

Having verified that $f(f^{-1}(x)) = x$, the next crucial step is to check if $f^{-1}(f(x)) = x$. This involves substituting the original function $f(x)$ into the inverse function $f^{-1}(x)$ and simplifying the expression. This step is equally important because, for two functions to be true inverses, they must “undo” each other regardless of the order in which they are applied. Let's start by recalling the functions:

• $f(x) = 5x^2 - 2$
• $f^{-1}(x) = \pm\sqrt{\frac{x+2}{5}}$

Now, substitute $f(x)$ into $f^{-1}(x)$:

$f^{-1}(f(x)) = \pm\sqrt{\frac{(5x^2 - 2) + 2}{5}}$

Simplify the expression inside the square root:

$f^{-1}(f(x)) = \pm\sqrt{\frac{5x^2}{5}}$

Further simplification yields:

$f^{-1}(f(x)) = \pm\sqrt{x^2}$

$f^{-1}(f(x)) = \pm x$

Here, we encounter a critical point. The result is $\pm x$, not simply x. This indicates that $f^{-1}(f(x))$ does not always return the original input x, especially when considering the negative root. This is because the original function $f(x) = 5x^2 - 2$ is not one-to-one over its entire domain (it fails the horizontal line test). The presence of the $x^2$ term means that both positive and negative values of x will yield the same output, which complicates the inverse relationship. The $\pm$ sign in the result highlights that the inverse function, as defined, returns both positive and negative roots, which is not a unique inverse. For $f^{-1}(x)$ to be a true inverse, it should only return one value for each input. This discrepancy means that while $f(f^{-1}(x)) = x$, the condition $f^{-1}(f(x)) = x$ is not fully satisfied due to the $\pm$ sign. Therefore, we can conclude that $f(x)$ and $f^{-1}(x)$ are not inverses over their entire domains. This analysis demonstrates the importance of verifying both compositions, $f(f^{-1}(x))$ and $f^{-1}(f(x))$, to definitively establish an inverse relationship between two functions.

Conclusion

In conclusion, to determine if $f(x) = 5x^2 - 2$ and $f^{-1}(x) = \pm\sqrt{\frac{x+2}{5}}$ are inverse functions, we must verify both $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. We found that while $f(f^{-1}(x))$ simplifies to x, $f^{-1}(f(x))$ results in $\pm x$. This discrepancy, where the inverse composition does not uniquely return the original input, indicates that the given functions are not inverses of each other over their entire domains. The issue arises because the original function $f(x) = 5x^2 - 2$ is not one-to-one, meaning it does not have a unique inverse across its entire domain. The square in the function causes both positive and negative values of x to map to the same y value, leading to the ambiguity in the inverse. This exploration highlights the importance of thoroughly checking both compositions when verifying inverse functions. It's not sufficient for just one composition to satisfy the inverse property; both must hold true. Furthermore, it illustrates that not all functions have inverses, especially if they are not one-to-one. Understanding these nuances is crucial in advanced mathematical studies, particularly in calculus and analysis, where inverse functions play a significant role. The process of verifying inverse functions involves careful algebraic manipulation and a clear understanding of the definitions. By meticulously working through each step, we can accurately determine whether two functions are indeed inverses and gain a deeper insight into the properties of mathematical functions. For further reading on inverse functions, you can visit resources like Khan Academy's article on Inverse Functions.