One-to-One Function Analysis F(-5) = -11 And F(2) = -12

Let's dive deep into one-to-one functions, also known as injective functions. Guys, these functions are super important in mathematics, especially when we're dealing with inverses and mappings. A one-to-one function has a unique property: each element in the range corresponds to exactly one element in the domain. Simply put, no two different inputs will produce the same output. Think of it like a perfect matching game – each input has its own special output, and no one else can have it!

To formally define it, a function $f$ is one-to-one if, for any $x_1$ and $x_2$ in the domain of $f$, if $f(x_1) = f(x_2)$, then $x_1 = x_2$. This definition is crucial because it gives us a direct way to check if a function is one-to-one. Another way to state this is: if $x_1 ≠ x_2$, then $f(x_1) ≠ f(x_2)$. This is just the contrapositive of the first definition, but it’s equally useful. Graphically, a function is one-to-one if it passes the horizontal line test, which means that no horizontal line intersects the graph of the function more than once. This visually confirms that each $y$-value corresponds to only one $x$-value.

The concept of one-to-one functions is foundational in many areas of mathematics. For example, when we talk about inverse functions, a function must be one-to-one to have an inverse. If a function isn’t one-to-one, we can’t simply reverse the mapping because we wouldn’t know where to send a particular output. One-to-one functions are also vital in cryptography, where unique mappings ensure that encrypted messages can be decrypted correctly. They pop up in set theory, calculus, and even linear algebra. So, grasping the essence of one-to-one functions is essential for anyone serious about math.

Key Properties of One-to-One Functions

Let’s talk about some essential characteristics that define one-to-one functions. A primary trait is the unique mapping, as we've mentioned. But let's dig deeper. If we have a function $f(x)$, and we know that $f(a) = b$, then for a one-to-one function, there’s no other $x$ value that will also map to $b$. This uniqueness is a game-changer in problem-solving.

Another crucial property is the existence of an inverse. A function $f$ has an inverse function, denoted as $f^{-1}$, if and only if $f$ is one-to-one. The inverse function essentially “undoes” what the original function did. If $f(a) = b$, then $f^{-1}(b) = a$. This inverse relationship is only possible because one-to-one functions ensure a clear, reversible mapping. Without this one-to-one property, we’d run into ambiguity when trying to reverse the process.

Moreover, one-to-one functions behave predictably with composition. If both $f$ and $g$ are one-to-one, then their composition, $f(g(x))$, is also one-to-one. This is significant when dealing with complex functions built from simpler ones. Knowing that one-to-one properties are preserved under composition simplifies many mathematical arguments and proofs. The domain and range of a one-to-one function and its inverse are also closely related. The domain of $f$ becomes the range of $f^{-1}$, and the range of $f$ becomes the domain of $f^{-1}$. This switch is crucial for understanding the behavior of inverse functions and their applicability.

Examples of One-to-One Functions

To solidify our understanding, let's look at some concrete examples of one-to-one functions. A classic example is the linear function $f(x) = mx + b$, where $m ≠ 0$. Why? Because for any two different $x$ values, say $x_1$ and $x_2$, the outputs $f(x_1)$ and $f(x_2)$ will also be different. The slope $m$ ensures that the function is strictly increasing or decreasing, which guarantees one-to-one behavior. You can visualize this as a straight line that either goes uphill or downhill without ever leveling off or turning back on itself.

Another great example is the exponential function, like $f(x) = a^x$, where $a > 0$ and $a ≠ 1$. Exponential functions are always either increasing or decreasing, so they pass the horizontal line test with flying colors. No matter what horizontal line you draw, it will only intersect the graph of an exponential function at most once. This makes exponential functions incredibly useful in various applications, from modeling population growth to describing radioactive decay.

Now, let's contrast these with functions that aren't one-to-one. A simple example is the quadratic function $f(x) = x^2$. This function fails the horizontal line test because, for any positive $y$ value, there are two $x$ values (one positive and one negative) that produce the same output. For instance, both $x = 2$ and $x = -2$ give you $f(x) = 4$. So, the quadratic function is a clear illustration of a function that is not one-to-one.

Analyzing the Given Function $f$

Okay, let's get to the heart of the matter. We're given a function $f$ that is one-to-one, and we know two specific values: $f(-5) = -11$ and $f(2) = -12$. These two points are our anchors, and the fact that $f$ is one-to-one is our guiding principle. Guys, this is key information, so let's see how we can use it.

Using the One-to-One Property

The most direct implication of $f$ being one-to-one is that if we input different values, we get different outputs. We already see this in action with our given points. Since $-5$ and $2$ are distinct inputs, the outputs $f(-5) = -11$ and $f(2) = -12$ are also distinct. This might seem like a no-brainer, but it’s fundamental to the definition of a one-to-one function. If we were to find another input, say $x$, such that $f(x) = -11$, we could immediately conclude that $x$ must be $-5$. Similarly, if $f(x) = -12$, then $x$ must be $2$.

This unique mapping allows us to make definite statements about the function's behavior. If we ever encounter an equation like $f(x) = f(y)$, we can confidently say that $x = y$. This is a powerful tool in simplifying expressions and solving equations involving one-to-one functions. It essentially means we can