Simplify G(x+1) For G(x) = -5x^2 + 3

Hey everyone! Today, we're diving deep into the world of functions, and specifically, we're going to tackle a common yet super important concept: evaluating a function at a different expression. Our main keyword here is, of course, finding g(x+1). This might sound a little tricky at first, but trust me, guys, once you break it down, it's as easy as pie. We're given a function, $g(x) = -5x^2 + 3$, and our mission, should we choose to accept it, is to figure out what $g(x+1)$ looks like. The key here is to understand function notation. Remember, when you see $g(x)$, it's like a placeholder. Whatever is inside those parentheses is what gets substituted into the function wherever you see an 'x'. So, if we have $g(x+1)$, that means the entire expression $(x+1)$ is going to replace every single 'x' in our original function definition. We're not just adding 1 to the result of $g(x)$; we're actually plugging in a whole new input. This is a fundamental skill in algebra, and mastering it will unlock a lot of other cool math concepts. We'll go through this step-by-step, ensuring we handle the expansion and simplification correctly. Get ready to flex those algebraic muscles, because by the end of this, you'll be a pro at evaluating functions with new inputs!

Understanding Function Notation and Substitution

Alright, let's get down to business with finding g(x+1). The function we're working with is $g(x) = -5x^2 + 3$. Now, the golden rule of function notation is this: whatever is inside the parentheses after the function name is your input. You take that input and substitute it everywhere you see the variable (in this case, 'x') in the function's definition. Think of the function $g$ as a machine. You put something in (like 'x'), and it performs a specific operation on it (-5 times the square of the input, plus 3) and gives you an output. Today, we're not putting a simple 'x' into the machine; we're putting the entire expression $(x+1)$ in. So, when we write $g(x+1)$, we are telling the 'g' machine to take $(x+1)$ and do the following: square it, multiply the result by -5, and then add 3. It’s like giving the machine a whole new recipe to follow! The most common mistake people make here is thinking they should calculate $g(x)$ first and then add 1. That's not what $g(x+1)$ means. It means replace 'x' with $(x+1)$ before you do any calculations. We'll need to be careful with the algebra, especially when squaring the term $(x+1)$. This is where the binomial expansion comes into play. Remember $(a+b)^2 = a^2 + 2ab + b^2$? We'll be using that exact pattern. So, our first step is to take the original function $g(x) = -5x^2 + 3$ and literally swap out every 'x' with $(x+1)$. This gives us $g(x+1) = -5(x+1)^2 + 3$. See? We've replaced the 'x' inside the square with $(x+1)$. Now, the real work begins: simplifying this expression. We need to get rid of those parentheses and combine any like terms to present our answer in its simplest form, as the prompt requested. This involves expanding the squared term and then distributing the -5. Keep your eyes peeled for errors, especially with signs – they can be sneaky! We're aiming for a simplified polynomial expression, free of any parentheses.

Step-by-Step Simplification of g(x+1)

Alright, guys, let's get our hands dirty and simplify $g(x+1)$ step-by-step. We've already established that $g(x+1) = -5(x+1)^2 + 3$. The first major hurdle is dealing with that squared term, $(x+1)^2$. This is where many people stumble, so let's be super careful. Remember the binomial expansion formula: $(a+b)^2 = a^2 + 2ab + b^2$. In our case, $a = x$ and $b = 1$. So, $(x+1)^2$ expands to $x^2 + 2(x)(1) + 1^2$, which simplifies to $x^2 + 2x + 1$. Easy peasy, right? Now, we substitute this expanded form back into our expression for $g(x+1)$:

$g(x+1) = -5(x^2 + 2x + 1) + 3$

Next up, we need to distribute the $-5$ to each term inside the parentheses. This is another critical step where signs can get mixed up. Multiplying $-5$ by $x^2$ gives us $-5x^2$. Multiplying $-5$ by $2x$ gives us $-10x$. And multiplying $-5$ by $1$ gives us $-5$. So, the expression now becomes:

$g(x+1) = -5x^2 - 10x - 5 + 3$

We're almost there! The final step in simplifying is to combine any like terms. In this expression, the only like terms are the constants: $-5$ and $+3$. Adding these together, we get $-5 + 3 = -2$.

So, our fully simplified expression for $g(x+1)$ is:

$g(x+1) = -5x^2 - 10x - 2$

And there you have it! We've successfully found $g(x+1)$ and simplified it without any parentheses. The process involved understanding function notation, careful substitution, applying the binomial expansion, distributing a coefficient, and combining like terms. Each step is crucial for arriving at the correct, simplified answer. Make sure you practice this with different functions and different inputs to really nail it. It's all about methodical execution and paying attention to the details, especially those pesky signs!

Why Simplifying is Essential

Now, you might be asking yourselves, "Why do we even bother with all this simplifying stuff?" That's a fair question, guys! The main reason simplifying expressions like g(x+1) is so important boils down to clarity and further analysis. When we have an expression like $-5(x+1)^2 + 3$, it tells us how to calculate the value for a given 'x' in a very direct way: add 1 to x, square the result, multiply by -5, and then add 3. It reflects the original function's structure applied to the new input. However, the simplified form, $-5x^2 - 10x - 2$, presents the function in its standard polynomial form. This form is incredibly useful for a number of reasons. Firstly, it makes it much easier to compare different functions or different forms of the same function. If you were asked to compare $g(x+1)$ with, say, $h(x) = -5x^2 - 10x - 2$, you could immediately see they are identical. Secondly, the simplified polynomial form is the standard for many advanced mathematical concepts. For instance, when you learn about derivatives in calculus, you'll want your function in this expanded polynomial form to easily apply the power rule. Finding the rate of change of $g(x+1)$ would be much simpler starting from $-5x^2 - 10x - 2$ than from $-5(x+1)^2 + 3$. Moreover, this simplified form helps in graphing the function. While the original form reveals transformations relative to $g(x)$ (like horizontal shifts), the standard polynomial form can be directly used in graphing calculators or software, or for analyzing intercepts and end behavior. It’s also less prone to calculation errors when evaluating for specific values. If you needed to find $g(3+1)$, you could plug $x=3$ into $-5(3+1)^2+3$ or into $-5(3)^2 - 10(3) - 2$. The latter often involves simpler arithmetic operations. So, while the process of simplification requires careful algebraic manipulation, the resulting form offers significant advantages for understanding, comparing, and further analyzing the function. It's all about making mathematics more accessible and manageable for deeper study and application. Keep practicing that simplification; it pays off big time!

Common Pitfalls When Finding g(x+1)

Alright, let's talk about the common mistakes people make when finding g(x+1). We want to make sure you guys avoid these traps so you can nail this every time! The first and perhaps the most frequent pitfall is misunderstanding function notation. As we stressed before, $g(x+1)$ means replacing every 'x' in $g(x)$ with the entire expression $(x+1)$. A big error is to calculate $g(x)$ first and then add 1, like $-5x^2 + 3 + 1$. This gives you $-5x^2 + 4$, which is totally different and incorrect. Always substitute first! The second major area where errors creep in is during the expansion of $(x+1)^2$. Many students mistakenly think $(x+1)^2 = x^2 + 1^2 = x^2 + 1$. This is a critical mistake, guys! Remember, $(x+1)^2$ means $(x+1)(x+1)$. You must use the distributive property (or FOIL method) or the binomial expansion formula: $(x+1)(x+1) = x(x) + x(1) + 1(x) + 1(1) = x^2 + x + x + 1 = x^2 + 2x + 1$. Forgetting the middle term, $2x$, is a very common oversight. Another common slip-up occurs during the distribution of the coefficient, in our case, $-5$. When you multiply $-5$ by $(x^2 + 2x + 1)$, you need to multiply it by each term inside the parentheses. A mistake here might be only multiplying $-5$ by the $x^2$ term, leading to $-5x^2 + 2x + 1$, or misapplying signs. For example, $-5 imes 2x$ should be $-10x$, not $+10x$. Lastly, even after getting the terms right, there's the potential for error when combining like terms. In our final step, we had $-5$ and $+3$. If you accidentally calculate this as $-8$ or $-53$, you'll end up with the wrong constant term. It should be $-5 + 3 = -2$. So, to recap the major pitfalls: 1. Incorrectly interpreting $g(x+1)$ as $g(x) + 1$. 2. Failing to correctly expand $(x+1)^2$ (missing the $2x$ term). 3. Errors in distributing the coefficient (especially with signs). 4. Mistakes when combining the constant terms. By being aware of these common traps and double-checking your work at each stage, you can significantly reduce the chances of making errors and ensure you arrive at the correct simplified expression for $g(x+1)$. Always take your time and be meticulous!

Conclusion

So, there you have it! We've successfully navigated the process of finding g(x+1) for the function $g(x) = -5x^2 + 3$. By carefully substituting $(x+1)$ for every $x$, expanding the binomial $(x+1)^2$ correctly, distributing the coefficient $-5$, and finally combining like terms, we arrived at the simplified expression: $g(x+1) = -5x^2 - 10x - 2$. Remember, the key takeaways are to always substitute first before performing any operations, to be meticulous with algebraic expansions like squaring binomials, and to pay close attention to signs during distribution and simplification. This skill of evaluating functions at different expressions is fundamental in mathematics and will serve you well as you tackle more complex problems in algebra and beyond. Keep practicing, stay curious, and don't be afraid to break down problems into smaller, manageable steps. You've got this!