Range Of Composite Functions U(v(x)) Explained Step-by-Step

Hey guys! Today, we're diving into the fascinating world of composite functions and their ranges. We've got a fun problem on our hands: Given the functions $u(x) = -2x^2 + 3$ and $v(x) = \frac{1}{x}$, we want to figure out the range of the composite function $(u \circ v)(x)$, which is essentially $u(v(x))$. This means we're plugging the function $v(x)$ into the function $u(x)$. Buckle up, because this is going to be an exciting mathematical journey!

Understanding Composite Functions

Before we jump into the specifics, let's make sure we're all on the same page about composite functions. A composite function is simply a function that is formed by plugging one function into another. In our case, $(u \circ v)(x)$ means we first apply the function $v$ to $x$, and then we take the result and plug it into the function $u$. Think of it like a mathematical assembly line, where each function performs a specific operation on the input it receives. To truly grasp this, we'll break down the functions $u(x)$ and $v(x)$ individually before combining them.

Analyzing the Function v(x) = 1/x

First, let's consider the function $v(x) = \frac{1}{x}$. This is a simple reciprocal function, but it has some crucial characteristics that we need to be aware of. The domain of $v(x)$ is all real numbers except for $x = 0$, since division by zero is undefined. So, $x$ can be anything but 0. The range of $v(x)$ is also all real numbers except for 0. This means that the output of $v(x)$ can be any number except 0. As $x$ approaches infinity, $v(x)$ approaches 0, and as $x$ approaches 0, $v(x)$ approaches infinity (or negative infinity, depending on the direction). Understanding the behavior of $v(x)$ is critical because its range becomes the domain for the next function in our composition, $u(x)$. This interplay between domain and range is a core concept in understanding composite functions.

Analyzing the Function u(x) = -2x² + 3

Now, let's turn our attention to the function $u(x) = -2x^2 + 3$. This is a quadratic function, specifically a parabola that opens downwards due to the negative coefficient of the $x^2$ term. The vertex of this parabola represents the maximum value of the function. To find the vertex, we can complete the square or use the vertex formula. In this case, the vertex is at $(0, 3)$, which means the maximum value of $u(x)$ is 3. Since the parabola opens downwards, the range of $u(x)$ is all real numbers less than or equal to 3, or $(-\infty, 3]$. The behavior of this function is determined by the square term, which ensures that the output will never exceed 3. This limitation will have significant implications when we combine it with $v(x)$.

Composing the Functions: u(v(x))

Okay, now for the main event: let's find the composite function $(u \circ v)(x)$. This means we need to substitute $v(x)$ into $u(x)$. So, we replace the $x$ in $u(x)$ with $\frac{1}{x}$, giving us:

$u(v(x)) = -2(\frac{1}{x})^2 + 3 = -2(\frac{1}{x^2}) + 3$.

Simplifying this, we get:

$u(v(x)) = -\frac{2}{x^2} + 3$.

This new function, $u(v(x))$, is the key to our problem. To determine its range, we need to analyze how the operations within the function affect the possible output values. The $x^2$ in the denominator ensures that the term $\frac{2}{x^2}$ is always positive (or zero). The negative sign in front of the fraction flips the sign, making the term $-\frac{2}{x^2}$ always negative (or zero). Finally, adding 3 shifts the entire function up by 3 units. This step-by-step analysis is essential to unravel the function's behavior and determine its range.

Determining the Range of u(v(x))

Now, let's tackle the crucial question: what is the range of $u(v(x)) = -\frac{2}{x^2} + 3$? We know that $x$ cannot be 0 because it's in the denominator of $v(x)$. As $x$ gets very large (either positive or negative), $x^2$ also gets very large, and the fraction $\frac{2}{x^2}$ approaches 0. This means that $- \frac{2}{x^2}$ also approaches 0. So, as $x$ goes to infinity, $u(v(x))$ approaches 3. However, $u(v(x))$ will never actually equal 3 because $\frac{2}{x^2}$ will never be exactly 0.

On the other hand, as $x$ gets closer to 0, $x^2$ gets very small, and the fraction $\frac{2}{x^2}$ becomes very large. This means that $- \frac{2}{x^2}$ becomes a large negative number. Consequently, $u(v(x)) = -\frac{2}{x^2} + 3$ becomes a very large negative number, approaching negative infinity. Therefore, the range of $u(v(x))$ includes all real numbers less than 3, but not 3 itself. This careful analysis of the function's behavior near critical points (like $x$ approaching infinity and $x$ approaching 0) is crucial for accurately determining its range.

Putting it All Together

So, we've figured out that $u(v(x))$ can take on any value less than 3, but it can't actually reach 3. This means the range of $(u \circ v)(x)$ is $(-\infty, 3)$. Looking back at the given options, we can see that option C, $(-\infty, 3]$, is the correct answer. But hold on! Notice the subtle difference: our range is $(-\infty, 3)$, which means 3 is not included, while option C has a square bracket, meaning 3 is included. This discrepancy highlights the importance of careful notation and attention to detail in mathematics. We need to exclude 3 from the range because the term $-\frac{2}{x^2}$ will never be exactly zero.

Final Answer and Takeaways

Therefore, the correct range of $(u \circ v)(x)$ is actually $(-\infty, 3)$. This wasn't one of the provided answer choices, which means there might be an error in the question or the options provided. But hey, that's okay! The important thing is that we walked through the process together and understood how to find the range of a composite function. Remember, the key is to:

1. Understand the individual functions: Analyze their domains and ranges.
2. Compose the functions: Substitute the inner function into the outer function.
3. Analyze the composite function: Determine how its behavior changes as $x$ varies, especially near critical points like 0 and infinity.
4. Express the range: Use correct notation to indicate whether endpoints are included or excluded.

By mastering these steps, you'll be well-equipped to tackle any composite function problem that comes your way. Keep practicing, keep exploring, and keep having fun with math! Remember, even if the answer choices aren't quite right, the process of understanding and solving the problem is what truly matters. You guys got this!

Let's clarify the original question to avoid ambiguity. The core issue is finding the range of the composite function $u(v(x))$ given $u(x) = -2x^2 + 3$ and $v(x) = \frac{1}{x}$. A clearer way to phrase the question would be: What is the range of the function obtained by composing $u(x) = -2x^2 + 3$ with $v(x) = \frac{1}{x}$ to form $u(v(x))$? Express your answer in interval notation.

Finding the Range of Composite Functions A Step-by-Step Guide