Math Function Evaluation: $g(x) = -2x^2 + 3x - 5$

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Hey guys, let's dive into the awesome world of functions in mathematics! Today, we're going to tackle a specific task: evaluating a function for different input values. Specifically, we'll be working with the function $g(x) = -2x^2 + 3x - 5$. Think of a function like a machine; you put something in (an input value), and it gives you something out (an output value). Our job is to figure out what comes out when we put in $-2$, $0$, and $3$ into our function machine $g(x)$. This skill is super fundamental in algebra and pretty much all of math, so buckle up and let's get this done. We'll break down each input step-by-step so you can really get a handle on how this works.

Understanding Function Notation and Evaluation

Alright, let's get to the nitty-gritty of what it means to "evaluate a function." When we see something like $g(x) = -2x^2 + 3x - 5$, the $g(x)$ part is just a way of saying "the output of function $g$ when the input is $x$." So, if we want to find out what $g$ does to a specific number, say $2$, we replace every instance of $x$ in the expression $-2x^2 + 3x - 5$ with the number $2$. This process is called substitution. After substituting, we just follow the order of operations (PEMDAS/BODMAS – Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) to simplify and find our final output. It's like giving the function machine a specific ingredient and seeing what delicious mathematical dish it cooks up. The values we plug in for $x$ are called the independent variables or input values, and the results we get are called the dependent variables or output values. Understanding this relationship is key, as it forms the basis for graphing, solving equations, and so much more in mathematics. We'll be using this precise substitution method for each of our given inputs: $-2$, $0$, and $3$. This isn't just about getting an answer; it's about understanding the process of how functions transform numbers. So, pay close attention to how we substitute and simplify. It's a skill that will serve you well in all your future mathematical endeavors, from simple algebra problems to complex calculus theorems. Mastering function evaluation is like learning the alphabet before you can read a novel – it's essential for everything that follows.

Evaluating $g(x)$ for $x = -2$

Okay, team, let's start with our first input value: $x = -2$. We need to find $g(-2)$. Remember, we're taking our function $g(x) = -2x^2 + 3x - 5$ and swapping out every $x$ with $-2$. It's crucial to use parentheses when substituting, especially with negative numbers, to avoid sign errors. So, here we go:

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

Now, let's follow the order of operations. First, we handle the exponent: $(-2)^2$. Squaring a negative number always results in a positive number. So, $(-2)^2 = (-2) imes (-2) = 4$.

Our expression now looks like this:

$g(-2) = -2(4) + 3(-2) - 5$

Next, we perform the multiplications. We have $-2(4)$ and $3(-2)$.

$-2(4) = -8$

$3(-2) = -6$

Plugging these results back in, we get:

$g(-2) = -8 + (-6) - 5$

Which simplifies to:

$g(-2) = -8 - 6 - 5$

Finally, we perform the additions and subtractions from left to right.

$-8 - 6 = -14$

$-14 - 5 = -19$

So, when our input is $-2$, the output of the function $g(x)$ is $\bf{-19}$. That is, $g(-2) = -19$. Pretty neat, huh? This means the point $(-2, -19)$ lies on the graph of this function. Every time you see $x$, just think "insert $-2$ here" and then crunch the numbers. The parentheses are your best friend, especially when negative signs are involved, to ensure you're squaring the entire number (including its sign) and multiplying correctly. Without them, you might accidentally calculate $-2^2$ as $-(2^2) = -4$, which is incorrect. Here, $(-2)^2$ means the whole $-2$ is multiplied by itself, yielding a positive $4$. This distinction is absolutely vital in function evaluation. So, for $x = -2$, we've successfully found that $g(-2) = -19$. We took the input $-2$, ran it through the function machine, and out popped $-19$. Mission accomplished for this value!

Evaluating $g(x)$ for $x = 0$

Moving on, let's evaluate our function $g(x) = -2x^2 + 3x - 5$ for the input value $x = 0$. This one is usually a bit simpler, guys, because zero tends to make things disappear!

We substitute $0$ for every $x$:

$g(0) = -2(0)^2 + 3(0) - 5$

Let's follow the order of operations again. First, the exponent: $(0)^2 = 0 imes 0 = 0$.

Now the expression is:

$g(0) = -2(0) + 3(0) - 5$

Next, the multiplications:

$-2(0) = 0$

$3(0) = 0$

Substituting these back:

$g(0) = 0 + 0 - 5$

And finally, the addition and subtraction:

$0 + 0 - 5 = -5$

So, for the input value $0$, the output is $\bf{-5}$. That means $g(0) = -5$. This result is often called the y-intercept of the function's graph, as it's the point where the graph crosses the y-axis. It's always a good idea to check the output for $x=0$ because it directly tells you the constant term of the polynomial (if it's a polynomial function like this one). In our case, the constant term is $-5$, and that's exactly what we got as our output. This consistency reinforces our understanding. When dealing with functions, especially polynomials, the value of the function at $x=0$ is frequently a point of interest, as it represents the starting value or baseline before any other operations associated with $x$ take effect. It's the value of the function when the independent variable is zero, hence the term y-intercept. This makes evaluating at zero a quick way to find a key feature of the function's graph. So, for our function $g(x)$, the point $(0, -5)$ is on its graph. Easy peasy!

Evaluating $g(x)$ for $x = 3$

Last but not least, let's tackle the input value $x = 3$. We're looking for $g(3)$ using our trusty function $g(x) = -2x^2 + 3x - 5$.

Substitute $3$ for every $x$:

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

Order of operations time! First, the exponent: $(3)^2 = 3 imes 3 = 9$.

Now our expression is:

$g(3) = -2(9) + 3(3) - 5$

Next, the multiplications:

$-2(9) = -18$

$3(3) = 9$

Plugging these back in:

$g(3) = -18 + 9 - 5$

Finally, the addition and subtraction, working from left to right:

$-18 + 9 = -9$

$-9 - 5 = -14$

So, when the input is $3$, the output of the function $g(x)$ is $\bf{-14}$. That is, $g(3) = -14$. This tells us that the point $(3, -14)$ is also on the graph of our function. Notice how the results for $x = -2$ and $x = 3$ are different, which is expected since they are different inputs. The squaring operation ($x^2$) often causes the outputs for positive and negative inputs of the same magnitude to be different, especially when combined with other terms. In this case, even though $3$ is positive, the negative coefficient of the $x^2$ term ($-2$) plays a significant role in determining the final output. Evaluating functions for various inputs like these helps us understand the behavior of the function across different parts of its domain. For a quadratic function like $g(x)$, we know its graph is a parabola, and these points help us sketch it and understand its shape and position. We've now successfully evaluated $g(x)$ for all three required input values, demonstrating a comprehensive understanding of the evaluation process. Keep practicing this, and you'll become a function evaluation pro in no time!

Summary of Results

So, to wrap things up, we've successfully evaluated the function $g(x) = -2x^2 + 3x - 5$ for the specific input values of $-2$, $0$, and $3$. It's always good practice to summarize these findings so you have a clear overview.

• For the input $x = -2$, the output is $g(-2) = \bf{-19}$.
• For the input $x = 0$, the output is $g(0) = \bf{-5}$.
• For the input $x = 3$, the output is $g(3) = \bf{-14}$.

These three points, $(-2, -19)$, $(0, -5)$, and $(3, -14)$, are all points that lie on the graph of the function $g(x)$. This process of plugging in values and calculating the outputs is fundamental to understanding how functions work. Whether you're studying algebra, calculus, or any other branch of mathematics, mastering function evaluation is a crucial step. Keep practicing these steps, pay attention to the order of operations, and don't forget those parentheses, especially with negative numbers, and you'll be golden. Keep exploring the amazing world of math, guys! It's full of fascinating patterns and connections waiting to be discovered.