Domain & Range: Comparing F(x), G(x), And H(x)

Hey guys! Let's dive into a fun math problem where we're going to compare the domain and range of three different functions: f(x) = 3x², g(x) = 1/(3x), and h(x) = 3x. We need to figure out which statements accurately describe how these functions behave. This involves understanding what inputs (domain) and outputs (range) are possible for each function. So, let's break it down step by step and make sure we really get what's going on here.

Understanding Domain and Range

Before we jump into the specifics, let's make sure we're all on the same page about domain and range. Think of a function like a machine: you put something in (the input, or x), and it spits something out (the output, or y). The domain is the set of all possible things you can put into the machine without breaking it – basically, all the valid x values. The range is the set of all possible things that the machine can spit out – all the possible y values.

For example, if we have a simple function like h(x) = 3x, we can put in any real number for x, and we'll get a real number out. So, the domain and range are both all real numbers. But what if we have a function like g(x) = 1/(3x)? Here, we can't put in x = 0, because that would make the denominator zero, and dividing by zero is a big no-no in math. So, the domain is all real numbers except 0. We'll explore the ranges of these functions more specifically in a bit.

Understanding these basic concepts is crucial because they form the foundation for analyzing more complex functions. We need to identify any restrictions on the input values (domain) and then determine the potential output values (range) based on the function's behavior. Now that we have this clear understanding, let's analyze each function given in the problem and pinpoint their unique characteristics.

Analyzing f(x) = 3x²

Let's start with the function f(x) = 3x². The first thing we want to think about is the domain: what values of x can we plug into this function? Well, we can square any real number, and then multiply it by 3, so there are no restrictions on x. The domain of f(x) is all real numbers, which we can write as (-∞, ∞).

Now, let's think about the range. Since we're squaring x, the result will always be non-negative (either zero or positive). When we multiply by 3, it stays non-negative. The smallest value we can get is 0 (when x = 0), and the function can get infinitely large as x gets bigger in either the positive or negative direction. Therefore, the range of f(x) is [0, ∞). This means the function's output will always be zero or a positive number. The squared nature of the function restricts the range to non-negative values, a key characteristic to remember.

The graph of f(x) = 3x² is a parabola that opens upwards. This visual representation reinforces our understanding of the range: the lowest point of the parabola is at y = 0, and it extends upwards indefinitely. Understanding the shape of the graph can often provide a quick visual check on our algebraic analysis of domain and range.

Analyzing g(x) = 1/(3x)

Next up, we have g(x) = 1/(3x). For the domain, we need to be careful about the denominator. We can't divide by zero, so 3x cannot be zero. This means x cannot be zero. So, the domain of g(x) is all real numbers except 0, which we can write as (-∞, 0) ∪ (0, ∞).

Now for the range. As x gets very large (positive or negative), the fraction 1/(3x) gets very close to zero, but it never actually reaches zero. Also, as x gets close to zero, the fraction becomes very large (positive or negative). This means the function can take on any value except 0. So, the range of g(x) is also all real numbers except 0, or (-∞, 0) ∪ (0, ∞). This reciprocal function exhibits a unique behavior, approaching but never reaching zero, a crucial aspect to consider.

The graph of g(x) = 1/(3x) is a hyperbola with two separate branches. It gets closer and closer to the x-axis (y = 0) as x goes to infinity or negative infinity, and it gets closer and closer to the y-axis (x = 0) as x approaches zero. This visual representation perfectly illustrates why both the domain and range exclude zero.

Analyzing h(x) = 3x

Finally, let's look at h(x) = 3x. This is a simple linear function. For the domain, we can plug in any real number for x, so the domain is all real numbers, or (-∞, ∞).

For the range, as x varies over all real numbers, 3x also varies over all real numbers. There are no restrictions on the output. So, the range of h(x) is also all real numbers, or (-∞, ∞). This linear function provides a straightforward example where every input produces a unique output, and all real numbers are attainable.

The graph of h(x) = 3x is a straight line passing through the origin. The constant slope of 3 ensures that the function increases (or decreases for negative x values) uniformly across its entire domain, covering all possible y-values in the process. This linear behavior is a fundamental concept in function analysis.

Comparing Domains and Ranges

Now that we've analyzed each function individually, let's compare their domains and ranges to answer the question. We found that:

• f(x) = 3x² has a domain of (-∞, ∞) and a range of [0, ∞).
• g(x) = 1/(3x) has a domain of (-∞, 0) ∪ (0, ∞) and a range of (-∞, 0) ∪ (0, ∞).
• h(x) = 3x has a domain of (-∞, ∞) and a range of (-∞, ∞).

Let's look at some statements based on these findings. We need to identify which two statements accurately compare the functions.

Statement 1: “All of the functions have a unique range.” This is incorrect. f(x) has a range of [0, ∞), g(x) has a range of (-∞, 0) ∪ (0, ∞), and h(x) has a range of (-∞, ∞). These ranges are different.

Statement 2: “The range of all three functions is...” (The statement is incomplete in the original question, but we can infer what it might be referring to). Given our analysis, there is no single range that fits all three functions.

By carefully comparing the domains and ranges we've identified, we can accurately evaluate statements and select the ones that truly reflect the functions' characteristics. This process of breaking down functions, analyzing their behaviors, and then comparing them is a core skill in mathematics.

Selecting the Correct Statements

Based on our analysis, we can now address the original question and select the two correct statements. The key is to carefully review the characteristics of each function's domain and range.

After thoroughly examining the domains and ranges, we can identify the statements that are factually accurate. Remember, the goal is to choose the statements that best describe the behavior and limitations of each function. The domain and range provide a complete picture of what a function can do, making them essential concepts in mathematical analysis.

By understanding how different functions behave, we can predict their outputs for any given input and compare their characteristics effectively. This is not just a mathematical exercise but a powerful tool for problem-solving in various fields. So, let’s confidently select the statements that correctly compare these functions!

I hope this helps you guys understand how to analyze and compare the domains and ranges of functions! It's a fundamental concept in math, and getting comfortable with it will make a lot of other topics easier. Keep practicing, and you'll become a pro in no time!