Find The Function Where X=2 Maps To 32
Hey guys! Let's dive into a fun math problem today. We've got a question that asks us to figure out which function, from a given list, will output 32 when we plug in 2 for x. It's like a little detective game, and we're the detectives! We have these functions to work with:
- h(x) = 3x
- f(x) = 2x - 62
- f(x) = -3x^2 - 4
- g(x) = 4(x + 3)^2 - 68
Our mission, should we choose to accept it (and we do!), is to test each function and see which one gives us the magic number 32 when we substitute x with 2. So, grab your calculators, put on your thinking caps, and let's get started!
Testing the Functions
Okay, let’s methodically go through each function and see what happens when we substitute x with 2. This is where the rubber meets the road, guys! We're not just guessing here; we're using good ol' mathematical principles to find our answer. Remember, the goal is to find the function that spits out 32 when we feed it the number 2.
Function h(x) = 3x
First up, we have h(x) = 3x. This is a straightforward linear function. To see if this is our winner, we replace x with 2:
h(2) = 3 * 2 = 6
Well, that's not 32. So, we can cross this one off our list. It's essential to be this precise, guys. Math isn't about close enough; it's about being exact. We need that perfect match of 32, and 6 just doesn't cut it. Let's move on to the next contender!
Function f(x) = 2x - 62
Next, we're looking at f(x) = 2x - 62. This is another linear function, but this time with a subtraction element. Let's plug in 2 for x and see what we get:
f(2) = 2 * 2 - 62 = 4 - 62 = -58
Nope, -58 isn't 32 either. We're not even close! This highlights the importance of paying attention to all parts of the equation. That -62 really throws things off, doesn't it? Don't get discouraged, though. We've still got two more functions to test, and one of them has to be the right one. Right?
Function f(x) = -3x^2 - 4
Now we have a quadratic function: f(x) = -3x^2 - 4. This one looks a bit more interesting because of the x squared term. Let's see what happens when we substitute:
f(2) = -3 * (2^2) - 4 = -3 * 4 - 4 = -12 - 4 = -16
Still not 32! We got -16 this time, which is closer than -58, but still not our target. It's crucial to remember the order of operations here (PEMDAS/BODMAS). We square the 2 first, then multiply by -3, and finally subtract 4. Each step is important! Let's keep going; we're down to our last function.
Function g(x) = 4(x + 3)^2 - 68
Finally, we have g(x) = 4(x + 3)^2 - 68. This function looks the most complex of the lot, with parentheses, squaring, multiplication, and subtraction. But don't let that intimidate you! We'll tackle it step by step. Let's plug in 2 for x:
g(2) = 4(2 + 3)^2 - 68 = 4(5)^2 - 68 = 4 * 25 - 68 = 100 - 68 = 32
Eureka! We found it! g(2) = 32. That means the function g(x) = 4(x + 3)^2 - 68 is the one that maps x = 2 to 32. It took some work, but we got there by being systematic and careful with our calculations.
Why This Matters: Function Evaluation
You might be thinking, "Okay, we found the function, but why does this matter?" Well, guys, this exercise highlights a fundamental concept in mathematics: function evaluation. Function evaluation is the process of finding the value of a function at a specific input. In simpler terms, it's like a machine: you put something in (x in this case), and the function spits something else out (the value of the function at x). Understanding this is crucial for all sorts of math and science applications.
Think about it: functions are used to model real-world relationships all the time. For example, a function might describe the trajectory of a ball you throw, the growth of a bacteria population, or the price of a stock over time. Being able to evaluate these functions at specific points allows us to make predictions and understand these relationships better. So, yeah, it's pretty important stuff!
Moreover, function evaluation is a building block for more advanced topics in calculus and beyond. When you start dealing with derivatives and integrals, you'll be evaluating functions left and right. So, mastering this skill now will set you up for success later on. Trust me on this one!
Common Mistakes to Avoid
Now that we've solved the problem, let's talk about some common pitfalls people encounter when evaluating functions. Knowing these mistakes can help you avoid them and ensure you get the correct answer every time. Nobody wants to make silly errors, right?
Order of Operations
The biggest culprit, by far, is messing up the order of operations. Remember PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). If you don't follow this order, you're almost guaranteed to get the wrong answer. We saw this in action when evaluating the quadratic function. If you subtract 4 before squaring and multiplying, you'll end up with a completely different result.
Incorrect Substitution
Another common mistake is substituting the value of x incorrectly. This might sound simple, but it's easy to do, especially when the function is complex or you're working quickly. Double-check that you've replaced every instance of x with the correct value. A small slip-up here can throw off the entire calculation. It's like a typo in code; it can cause the whole program to crash!
Sign Errors
Sign errors are the bane of many math students' existence. A misplaced negative sign can completely change the answer. Be extra careful when dealing with negative numbers, especially when squaring or multiplying. It's a good practice to rewrite the expression with the substituted values, paying close attention to the signs. A little extra caution here can save you a lot of grief.
Arithmetic Errors
Finally, simple arithmetic errors can trip you up. Whether it's adding, subtracting, multiplying, or dividing, it's easy to make a mistake, especially under pressure. If you're working on a test or a timed assignment, take a moment to double-check your calculations. It's better to be accurate than fast. Using a calculator can help, but don't rely on it blindly. Always think about whether the answer makes sense in the context of the problem.
Practice Makes Perfect
So, what's the best way to avoid these mistakes? You guessed it: practice! The more you evaluate functions, the more comfortable and confident you'll become. Start with simple functions and gradually work your way up to more complex ones. Do plenty of examples, and don't be afraid to ask for help if you're struggling. Math is a skill, just like playing a musical instrument or learning a new language. It takes time and effort to master, but it's totally worth it in the end!
Conclusion
In conclusion, we successfully identified that the function g(x) = 4(x + 3)^2 - 68 maps x = 2 to 32. We did this by systematically evaluating each function and applying the correct order of operations. We also discussed the importance of function evaluation and common mistakes to avoid. Remember, guys, math is a journey, not a destination. Keep practicing, keep asking questions, and keep having fun with it!