Find G(4) If G(x) = 3x + 5: Step-by-Step Solution
Hey guys! Let's dive into a super common type of math problem you'll often see in algebra: evaluating functions. Specifically, we're going to tackle the question: If g(x) = 3x + 5, find g(4). Don't worry; it's much simpler than it sounds! We'll break it down step-by-step so you can confidently solve similar problems on your own. So, grab your pencil and paper, and let's get started!
Understanding the Problem
Before we jump into the solution, let's make sure we understand what the problem is asking. When you see something like "g(x) = 3x + 5", it's telling you that we have a function named "g" that takes an input "x", does some stuff to it (in this case, multiplies it by 3 and adds 5), and then gives you an output. The question "find g(4)" is asking: "What is the output of the function g when the input is 4?"
In simpler terms: Imagine 'g' is a machine. You feed it a number 'x', and it spits out a new number based on the rule '3x + 5'. We want to know what number the machine spits out when we feed it the number 4.
Breaking Down the Notation
- g(x): This is the function notation. 'g' is the name of the function, and 'x' is the input variable.
- 3x + 5: This is the rule that the function follows. It tells you what to do with the input 'x'.
- g(4): This means we're plugging in '4' for 'x' in the function.
Think of g(x) as a recipe. The x is the ingredient, and the 3x + 5 is the instruction on what to do with that ingredient. g(4) means you're using '4' as your ingredient and following the same instructions. Make sense? Let's move on to solving the problem!
Step-by-Step Solution
Now that we understand what the question is asking, let's solve it! Here’s how to find g(4) when g(x) = 3x + 5:
Step 1: Substitute
The first step is to replace the variable 'x' in the function g(x) with the number '4'. So, wherever you see 'x' in the expression '3x + 5', put a '4' instead. This gives us:
g(4) = 3(4) + 5
Remember: The parentheses mean multiplication. So, 3(4) means 3 times 4.
Step 2: Multiply
Next, we perform the multiplication: 3 times 4. This is a basic multiplication fact, and 3 multiplied by 4 equals 12. So now our equation looks like this:
g(4) = 12 + 5
Step 3: Add
Finally, we perform the addition. We add 12 and 5, which equals 17. Therefore:
g(4) = 17
And that's it! We've found that g(4) is equal to 17. Easy peasy, right?
Final Answer
The final answer is:
g(4) = 17
This means that when you input '4' into the function 'g', the output is '17'.
Practice Problems
Okay, now that you've seen how to solve this type of problem, let's try a couple of practice problems to make sure you've got it down. Here are two similar problems for you to try on your own:
- If f(x) = 2x - 1, find f(3).
- If h(x) = x/2 + 4, find h(10).
Take your time, follow the steps we outlined above, and see if you can find the correct answers. The answers are provided below, but try to solve them on your own first!
Answers to Practice Problems
- f(3) = 5
- h(10) = 9
Did you get them right? If so, great job! If not, don't worry. Go back and review the steps, and try again. Practice makes perfect!
Why is This Important?
You might be wondering, "Why is this even important? Where will I ever use this in real life?" Well, understanding functions and how to evaluate them is a fundamental concept in algebra and calculus. It's used in many different fields, including:
- Computer Science: Functions are used extensively in programming to create reusable blocks of code.
- Engineering: Engineers use functions to model and analyze systems, such as electrical circuits or mechanical structures.
- Economics: Economists use functions to model economic relationships, such as supply and demand.
- Data Science: Data scientists use functions to analyze and manipulate data.
So, even if you don't see a direct application right away, understanding functions will give you a solid foundation for future studies and career paths.
Common Mistakes to Avoid
When evaluating functions, there are a few common mistakes that students often make. Here are a few things to watch out for:
- Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS). Perform multiplication before addition or subtraction.
- Incorrect Substitution: Make sure you substitute the correct value for 'x'. Double-check that you're replacing 'x' with the given number.
- Arithmetic Errors: Be careful with your arithmetic. Simple addition, subtraction, multiplication, or division errors can lead to an incorrect answer.
By being aware of these common mistakes, you can avoid them and improve your accuracy.
Level Up Your Function Skills
Want to take your function skills to the next level? Here are a few ideas:
- Explore Different Types of Functions: We've only looked at linear functions in this example, but there are many other types of functions, such as quadratic, exponential, and trigonometric functions. Learning about these different types of functions will broaden your understanding of mathematics.
- Graph Functions: Graphing functions can help you visualize their behavior and understand their properties. There are many online tools that can help you graph functions.
- Solve More Complex Problems: Challenge yourself by solving more complex problems involving functions. Look for problems that require you to combine multiple concepts or apply your knowledge in new ways.
Conclusion
So, there you have it! We've walked through how to find g(4) if g(x) = 3x + 5, step by step. Remember, the key is to substitute the given value for 'x' and then simplify the expression using the order of operations. With a little practice, you'll be evaluating functions like a pro in no time! Keep practicing, and don't be afraid to ask for help if you get stuck. You've got this!
Understanding how to evaluate functions is a crucial skill in algebra and beyond. By mastering this concept, you're setting yourself up for success in more advanced math courses and various real-world applications. So keep practicing, stay curious, and never stop learning! Now go out there and conquer those functions!
