Analisis Soal Fungsi Matematika: Pernyataan Yang Benar

Hey guys! Let's dive into this cool math problem! We're given two functions, $f(x)$ and $g(x)$, and a bunch of statements. Our mission? To figure out which statements are true. It's like a mathematical treasure hunt, and we're the explorers! Understanding functions is key here. Remember, a function takes an input (x) and spits out an output based on a specific rule. So, let's break down each part of this problem step by step to get those correct answers. This is a common type of question in mathematics, testing your understanding of function operations and evaluation. Ready to get started? Let’s do it!

Memahami Fungsi dan Operasi Aljabar

Okay, before we jump into the statements, let's refresh our memory on functions and basic algebra. Function basically takes an input and produces an output based on a specific rule. In this case, we have two functions: $f(x) = 10x - 1$ and $g(x) = 20 + 2x$. The "f(x)" and "g(x)" are just names for these functions, like nicknames. When you see something like $f(1)$, it means we're plugging in the value 1 for 'x' in the function $f(x)$. So, it’s all about substituting the value of 'x' into the equation. Besides, the math involves some basic operations like addition, subtraction, multiplication, and division. So, we'll need to remember how to handle these operations with algebraic expressions. Remember the order of operations (PEMDAS/BODMAS)? It's crucial! Parentheses/Brackets first, then Exponents/Orders, then Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right).

We need to evaluate the different operations applied to these functions. For instance, $f(x) + g(x)$ means we add the two functions together. We combine like terms. The same goes for subtraction, we subtract the function. For multiplication, we simply multiply the two function outputs. And for division, it’s similar, we divide. Also, remember to be careful with signs when you're adding and subtracting expressions. A misplaced minus sign can totally mess up your answer. Understanding these fundamental concepts is absolutely essential to tackling this problem. Let's start breaking down the statements to see what's true and false. It's like a detective case, and we have to examine each clue carefully. The basic operations on functions are tested in most mathematics exams.

Memeriksa Pernyataan Satu per Satu

Alright, let's put on our detective hats and examine each statement to see if they're true or false. This is where we apply the function operations we just reviewed. We'll start with statement (1): $f(x) + g(x) = 12x + 21$. To check this, we need to add the two functions: $f(x) + g(x) = (10x - 1) + (20 + 2x)$. Combine like terms which gives us $10x + 2x - 1 + 20 = 12x + 19$. So, statement (1) is incorrect because the result should be $12x + 19$, not $12x + 21$. Statement (2): $f(x) - g(x) = 8x - 21$. Let’s find the difference: $f(x) - g(x) = (10x - 1) - (20 + 2x)$. This simplifies to $10x - 2x - 1 - 20 = 8x - 21$. So, statement (2) is correct. Awesome! We're making progress.

Now, let's look at statement (3): $f(1) \times g(1) = 198$. Here, we need to evaluate the functions at $x = 1$. First, find $f(1) = 10(1) - 1 = 9$. Then, find $g(1) = 20 + 2(1) = 22$. Multiply them together: $f(1) \times g(1) = 9 \times 22 = 198$. Statement (3) is correct too! We're on fire! For the last statement, we need to find if (4) $f(1):g(1)=\frac{9}{21}$. We already know that $f(1) = 9$ and $g(1) = 22$. Thus, $f(1) : g(1) = \frac{9}{22}$. Comparing it to the option $\frac{9}{21}$, we see that statement (4) is incorrect. The key here is to carefully substitute the values and perform the calculations. Remember, even a small mistake can lead to the wrong answer. So, we've carefully evaluated each statement and found which ones are true. It's all about precision and a good understanding of the function operations.

Menentukan Jawaban yang Benar

Based on our step-by-step analysis, we know that statements (2) and (3) are correct, and statements (1) and (4) are not. So, the correct answer is the option that includes statements (2) and (3). The process involves understanding the functions, carrying out operations, and carefully evaluating the results. It is important to avoid common mistakes, such as errors in algebraic manipulation and misinterpreting the operations. With practice, you can easily master these concepts and solve function-based problems like a pro. This question requires a combination of algebraic manipulation, functional evaluation, and logical reasoning. So, the right answer is the one that includes the correct statements. Remember the steps to get the right answer: understand the question, identify the information given, plan your approach, solve carefully, and check your answer. Keep practicing, and you'll become a function master in no time! Also, try to solve similar problems to reinforce your understanding and improve your problem-solving skills.

In conclusion, understanding functions, performing algebraic operations, and carefully evaluating each statement are the keys to solving this problem. Keep up the great work, and you'll do great! We've successfully analyzed each statement and pinpointed the correct ones. Remember to practice regularly and stay sharp with your math skills, guys! Keep up the good work and happy solving! And remember, math is like a puzzle – the more you play, the better you get! The understanding of functions and their operations is a fundamental skill in mathematics and serves as a building block for more complex topics in calculus and other related fields. Keep practicing, and you'll ace these problems in no time!