Analyzing Functions: Find True Statements For F(x) & G(x)
Hey Plastik Magazine readers! Let's dive into some math today, specifically analyzing functions. We've got two functions defined for us: f(x) = 5 - 0.2x and g(x) = 0.2(x + 5). Our mission, should we choose to accept it, is to figure out which statements about these functions are actually true. We'll go through each option step-by-step, so you can follow along and understand the reasoning behind the answers. Think of it like a mathematical puzzle – fun, right? Let’s get started and see what we can uncover about these functions. Grab your thinking caps, and let's make some mathematical magic happen!
Evaluating Statements About f(x) and g(x)
Let's break down each statement and see if it holds up under scrutiny. It's like being a math detective, and each statement is a clue we need to investigate. We'll use our trusty functions, f(x) and g(x), as our tools to solve the mystery. Remember, f(x) = 5 - 0.2x and g(x) = 0.2(x + 5). We'll substitute values and do some calculations, just like plugging numbers into a machine to see what comes out. By carefully evaluating each statement, we can separate the true from the false and ace this problem. So, let's put on our detective hats and get to work!
A. f(3) > 0
The first statement we're tackling is f(3) > 0. What this means is, if we plug in 3 for x in the function f(x), will the result be greater than zero? Let’s find out! We'll substitute x with 3 in our function f(x) = 5 - 0.2x, which gives us f(3) = 5 - 0.2(3). Now, we just need to do the math. First, multiply 0.2 by 3, which equals 0.6. Then, subtract 0.6 from 5. This gives us f(3) = 5 - 0.6 = 4.4. So, f(3) is equal to 4.4. Is 4.4 greater than 0? You bet it is! Therefore, the statement f(3) > 0 is absolutely true. We've cracked our first clue! This shows us how substituting values into functions can help us understand their behavior. Let's keep going and see what the other statements reveal.
B. f(3) > 5
Next up, we have the statement f(3) > 5. We already know from our previous calculation that f(3) = 4.4. Now, we need to determine if 4.4 is greater than 5. Take a moment to think about it. Is 4.4 bigger than 5? Nope, it isn't! 4. 4 is actually less than 5. Therefore, the statement f(3) > 5 is definitely false. This is a good reminder that even though a value might be positive (as we saw in statement A), it doesn't necessarily mean it's greater than another specific number. We need to compare the values directly. So, we've successfully debunked another statement. Let's move on to the next one and see what it holds.
C. g(-1) = 0.8
Now, let's shift our focus to the function g(x) and the statement g(-1) = 0.8. This means we need to plug in -1 for x in the function g(x) = 0.2(x + 5) and see if we get 0.8 as a result. So, let's substitute: g(-1) = 0.2(-1 + 5). First, we need to simplify the expression inside the parentheses: -1 + 5 equals 4. So, now we have g(-1) = 0.2(4). Next, we multiply 0.2 by 4, which gives us 0.8. Therefore, g(-1) = 0.8. Guess what? The statement g(-1) = 0.8 is absolutely true! We've correctly evaluated the function at a specific point. This shows how important it is to follow the order of operations (parentheses first!) when working with functions. Let's keep our momentum going and check out the next statement.
D. g(-1) < f(-1)
Statement D introduces a comparison between our two functions: g(-1) < f(-1). We already know that g(-1) = 0.8 from our previous calculation. Now, we need to figure out what f(-1) is. Remember, f(x) = 5 - 0.2x, so let’s substitute x with -1: f(-1) = 5 - 0.2(-1). Multiplying -0.2 by -1 gives us 0.2. So, f(-1) = 5 + 0.2 = 5.2. Now we can compare: is g(-1), which is 0.8, less than f(-1), which is 5.2? Yes, it definitely is! 0.8 is smaller than 5.2. Therefore, the statement g(-1) < f(-1) is also true. We're on a roll here! This statement highlights the importance of comparing function values at the same input to understand their relative behavior. Only one statement left to go – let's see what it holds.
E. f(0) = g(0)
Our final statement compares the values of f(x) and g(x) when x is 0: f(0) = g(0). Let's start by finding f(0). Using the function f(x) = 5 - 0.2x, we substitute x with 0: f(0) = 5 - 0.2(0). Since anything multiplied by 0 is 0, we have f(0) = 5 - 0 = 5. Now, let's find g(0). Using the function g(x) = 0.2(x + 5), we substitute x with 0: g(0) = 0.2(0 + 5). Simplifying inside the parentheses, we get g(0) = 0.2(5). Multiplying 0.2 by 5 gives us 1. So, g(0) = 1. Now we compare: Is f(0), which is 5, equal to g(0), which is 1? No, they are not equal! Therefore, the statement f(0) = g(0) is false. We've reached the end of our mathematical investigation, and we've successfully determined the truthfulness of all the statements. High five!
Conclusion: True Statements Identified
Alright, guys, we've reached the end of our function analysis adventure! We carefully examined each statement, plugged in values, and did the calculations to figure out which ones are true. Let's recap our findings. We discovered that the following statements are true:
- A. f(3) > 0
- C. g(-1) = 0.8
- D. g(-1) < f(-1)
So, there you have it! We've successfully navigated the world of functions and inequalities. Remember, practice makes perfect, so keep exploring and experimenting with different functions. Math can be fun, especially when you break it down step by step. Until next time, keep those brains buzzing and stay curious!