Understanding The Range: A Deep Dive Into The Function F(x)

Hey math enthusiasts! Today, we're diving deep into the fascinating world of functions, specifically focusing on how to determine the range of a given function. We'll be breaking down the function $f(x)=\frac{3}{4}|x|-3$, and by the end of this, you'll have a crystal-clear understanding of its range. So, grab your pencils, open your minds, and let's get started!

What Exactly is the Range, Anyway?

Before we jump into the function itself, let's make sure we're all on the same page about what the range actually is. In simple terms, the range of a function is the set of all possible output values (also known as y-values) that the function can produce. Think of it like this: you input some x-values, and the function cranks out some y-values. The range is just a list of all the different y-values the function can spit out.

It's important to distinguish the range from the domain, which is the set of all possible input values (x-values) for which the function is defined. In most cases, unless there are specific restrictions (like division by zero or square roots of negative numbers), the domain of a function can often be all real numbers. However, the range can be quite different. Determining the range often involves considering the behavior of the function, especially its key components and transformations. Understanding the range is important for a complete understanding of how the function behaves. Consider it like understanding the function's capabilities. If you want to use the function to do some task, you need to know what outputs it can give. If you're building a machine, you need to understand its output capacity. The range gives you that insight. For example, if you were to solve the function for a certain value of y, you can only do so if it is within the range. The range is super crucial in a lot of mathematical applications, as well as real-world scenarios. We use this concept in everything from physics to computer science, making sure that what our functions can give us is exactly what we expect.

Deciphering the Function $f(x)=\frac{3}{4}|x|-3$: A Step-by-Step Breakdown

Alright, let's get our hands dirty with the function $f(x)=\frac{3}{4}|x|-3$. This function has a few key features that are going to help us determine its range. Specifically, it involves the absolute value of x and some simple arithmetic operations. Let's break it down step-by-step:

1. The Absolute Value $|x|$: The absolute value function takes any real number x and returns its non-negative value. For instance, $|3| = 3$ and $|-3| = 3$. This means that the output of $|x|$ is always going to be greater than or equal to zero. This is a crucial element of the entire function, because it will impact what outputs we can produce.

2. Multiplication by $\frac{3}{4}$: Next, the absolute value of x is multiplied by $\frac{3}{4}$. Multiplying by a positive number like $\frac{3}{4}$ doesn't change the fact that the output will be non-negative. It simply scales the output. Because the absolute value is always non-negative, the result of $\frac{3}{4}|x|$ will also always be greater than or equal to zero. It's like taking the non-negative number and multiplying it by a non-negative constant, and the product will also be non-negative. For example, $\frac{3}{4} * 0 = 0$, $\frac{3}{4} * 2 = 1.5$, $\frac{3}{4} * 4 = 3$, and so on. Note that multiplying by a fraction, in this case, a fraction that is less than 1 (but greater than 0), makes the product less than the original absolute value. This scales it down, which is also important for the final range.

3. Subtracting 3: Finally, we subtract 3 from the result of $\frac{3}{4}|x|$. This is where the magic happens. Since $\frac{3}{4}|x|$ is always greater than or equal to zero, subtracting 3 shifts the entire function downwards on the y-axis. The subtraction of 3 from the overall expression has the effect of translating the function, or shifting it down the coordinate plane. Think of it like this: the smallest value that $\frac{3}{4}|x|$ can be is 0. If we subtract 3 from that, we get -3. Because of this, we know that all the possible values of the output will always be greater than or equal to -3. The output will never be less than -3.

Unveiling the Range: Putting It All Together

Based on our step-by-step analysis, we can deduce the range of the function $f(x)=\frac{3}{4}|x|-3$.

• The absolute value function ensures that $|x|$ is always greater than or equal to zero.
• Multiplying by $\frac{3}{4}$ keeps the result non-negative.
• Subtracting 3 shifts the entire function down by 3 units on the y-axis.

Therefore, the smallest possible value that the function can produce is -3 (when $|x| = 0$). As x moves away from zero (in either the positive or negative direction), the value of $\frac{3}{4}|x|$ increases, and so does the value of the function. This means that the function can take on any value greater than or equal to -3. The range of the function is all real numbers greater than or equal to -3. So the range of this function is $y \geq -3$ or in interval notation $[-3, \infty)$.

Matching the Answer Choices

Now, let's see which of the answer choices matches our findings.

A. all real numbers: This is incorrect because the function has a lower bound.

B. all real numbers less than or equal to 3: This is incorrect because the range does not have an upper bound, but has a lower bound of -3.

C. all real numbers less than or equal to -3: This is incorrect because our findings conclude a range of all real numbers greater than or equal to -3.

D. all real numbers greater than or equal to -3: This is the correct answer! This aligns perfectly with our understanding of the function's behavior.

Conclusion: You've Got This!

Awesome work, everyone! We've successfully determined the range of $f(x)=\frac{3}{4}|x|-3$. Remember, the key is to break down the function into its components, understand how each component affects the output, and then put it all together. Keep practicing, and you'll become a range-finding master in no time! Keep in mind the characteristics of each component in the function, such as the absolute value, the scale, and the translation. You can conquer these types of problems if you remember these concepts. Understanding the range of a function is crucial in many areas of mathematics and its applications, so make sure to practice and hone your skills. Keep up the excellent work! You got this!