Calculate F(-1) * G(-1) For Two Functions

Welcome, math enthusiasts! Today, we're diving into a fundamental concept in algebra: evaluating functions and then combining their results. Specifically, we'll be working with two functions, $f(x) = x^2 + 1$ and $g(x) = 3x + 5$, and our goal is to find the value of $f(-1)g(-1)$. This means we need to first determine the value of $f(x)$ when $x$ is $-1$, and then determine the value of $g(x)$ when $x$ is $-1$. Once we have these two individual values, we'll multiply them together to get our final answer. It's a straightforward process, but it requires careful attention to detail, especially when dealing with negative numbers and exponents. Let's break it down step-by-step to ensure clarity and accuracy. Understanding how to manipulate and evaluate these algebraic expressions is a cornerstone of higher mathematics, from calculus to linear algebra, and it's a skill that will serve you well in various academic and professional fields. So, grab your thinking caps, and let's get started on this engaging mathematical journey. We'll explore not just the mechanics of the calculation but also the underlying principles that make these operations possible. Our exploration will be thorough, aiming to leave no stone unturned as we demystify the process of function evaluation and multiplication.

Understanding Function Notation and Evaluation

Before we jump into calculating $f(-1)g(-1)$, let's first clarify what function notation and evaluation actually mean. The expression $f(x) = x^2 + 1$ tells us that for any input value 'x', the function 'f' will perform a specific operation: it will square the input value and then add 1 to the result. Similarly, $g(x) = 3x + 5$ indicates that for any input 'x', the function 'g' will multiply the input by 3 and then add 5 to that product. The notation $f(-1)$ is a command to substitute $-1$ for every occurrence of $x$ in the definition of $f(x)$. The same logic applies to $g(-1)$. This is the core of what it means to evaluate a function at a specific point. It’s like plugging a specific number into a machine and seeing what output you get based on the machine's predefined rules. The power of functions lies in their ability to represent relationships and patterns. By using a variable like 'x', we can define a rule that works for any number, and then by evaluating at specific points, we can see how that rule behaves for particular inputs. This is a fundamental concept in algebra and is the basis for graphing, modeling real-world phenomena, and solving complex equations. The clarity and conciseness of function notation allow mathematicians to express intricate ideas in a compact and universally understood way. When we talk about $f(-1)g(-1)$, we are essentially asking for the product of the outputs of these two functions when the same input, $-1$, is fed into both. It’s a way to explore how different mathematical operations interact and how their results can be combined. This process is crucial for understanding composite functions, where the output of one function becomes the input of another, and for analyzing the behavior of functions in various contexts. Mastery of this skill sets the stage for more advanced mathematical explorations.

Step 1: Evaluating f(-1)

Our first task is to find the value of the function $f(x) = x^2 + 1$ when $x = -1$. To do this, we substitute $-1$ for every $x$ in the expression for $f(x)$. So, $f(-1)$ becomes $(-1)^2 + 1$. Now, we need to be careful with the exponent. Squaring a negative number means multiplying it by itself. Therefore, $(-1)^2 = (-1) imes (-1)$. When you multiply two negative numbers, the result is a positive number. So, $(-1) imes (-1) = 1$. After calculating the squared term, we add 1 to it: $1 + 1 = 2$. Thus, the value of $f(-1)$ is 2. It's essential to remember the order of operations (PEMDAS/BODMAS), which dictates that exponents should be calculated before addition. In this case, squaring $-1$ must be done before adding 1. A common mistake is to incorrectly interpret $(-1)^2$ as $-1^2$, which would yield $-1$. However, the parentheses clearly indicate that the entire term $-1$ is being squared. This distinction is critical in algebraic calculations. The function $f(x)=x^2+1$ describes a parabola that opens upwards, with its vertex at $(0,1)$. Evaluating $f(-1)$ tells us the y-coordinate of a point on this parabola when the x-coordinate is $-1$. This point is $(-1, 2)$. Understanding this geometric interpretation can sometimes aid in visualizing the results of function evaluations. The precision in handling negative signs and exponents is paramount in mathematics, as even a small error can lead to a significantly different outcome. This methodical approach ensures that we are building a solid foundation for more complex mathematical problem-solving.

Step 2: Evaluating g(-1)

Next, we turn our attention to the function $g(x) = 3x + 5$. We need to find the value of $g(x)$ when $x = -1$. Following the same procedure as before, we substitute $-1$ for every $x$ in the expression for $g(x)$. So, $g(-1)$ becomes $3(-1) + 5$. Here, we have a multiplication and an addition. According to the order of operations, multiplication comes before addition. So, first, we calculate $3 imes (-1)$. Multiplying a positive number by a negative number results in a negative number. Thus, $3 imes (-1) = -3$. Now, we add 5 to this result: $-3 + 5$. When adding a negative number to a positive number, we can think of it as subtracting the absolute value of the negative number from the positive number, or simply counting up from the negative number. Starting from $-3$ and adding 5 brings us to $2$. So, $-3 + 5 = 2$. Therefore, the value of $g(-1)$ is 2. The function $g(x) = 3x + 5$ represents a linear equation, describing a straight line with a slope of 3 and a y-intercept of 5. Evaluating $g(-1)$ gives us the y-value on this line when the x-value is $-1$. This corresponds to the point $(-1, 2)$ on the line. Notice that for this particular input value, both functions yield the same output. This is a coincidence that occurs in this specific problem but doesn't always happen when evaluating different functions at the same point. The careful application of arithmetic rules, especially with signed numbers, is crucial for accurate function evaluation. Each step in the process, from substitution to the final arithmetic, must be performed with precision.

Step 3: Calculating f(-1)g(-1)

We have successfully evaluated both functions at $x = -1$. We found that $f(-1) = 2$ and $g(-1) = 2$. The problem asks us to find the value of $f(-1)g(-1)$. This notation means we need to multiply the result of $f(-1)$ by the result of $g(-1)$. So, we will calculate $2 imes 2$. Multiplying 2 by 2 gives us 4. Therefore, $f(-1)g(-1) = 4$. This final step is the culmination of our evaluation process. We take the individual outputs we calculated and combine them through multiplication as specified by the problem. This demonstrates how different function values can be integrated to solve a larger problem. In scenarios involving multiple functions, understanding how to combine their outputs is a vital skill. For example, in physics, you might have a function describing the velocity of an object over time and another function describing its mass; you might need to multiply these to find momentum. In economics, you might have functions for supply and demand, and their interaction determines market price. The simple act of multiplication here signifies a relationship between the outputs of $f(x)$ and $g(x)$ at the point $x=-1$. The fact that both functions yield the same output for this specific input might seem interesting, and it highlights that different mathematical relationships can intersect or align at particular points. This final calculation is a direct answer to the posed question, confirming that the product of the function values at $x=-1$ is indeed 4. It's a concise and clear result derived from careful, step-by-step algebraic manipulation.

Conclusion and Further Exploration

In summary, we have successfully determined that for the functions $f(x) = x^2 + 1$ and $g(x) = 3x + 5$, the value of $f(-1)g(-1)$ is 4. This was achieved by first evaluating $f(-1)$, which resulted in 2, and then evaluating $g(-1)$, which also resulted in 2. Finally, we multiplied these two values together: $2 imes 2 = 4$. This exercise underscores the importance of understanding function notation, the order of operations, and the careful handling of negative numbers in algebraic computations. These foundational skills are crucial for tackling more complex mathematical challenges. The journey through evaluating functions can lead to many interesting avenues. For instance, you might explore what happens if you calculate $f(g(-1))$ or $g(f(-1))$, which are examples of composite functions. Understanding how to compose functions is a significant step in algebra and calculus. You could also explore the values of $f(x)$ and $g(x)$ for different input values of $x$ to see how their outputs change. Graphing these functions can provide a visual representation of their behavior and help you understand where they intersect or how their values relate at various points. For further exploration into the fascinating world of functions and their applications, I recommend checking out resources like Khan Academy's extensive library on algebra, which offers free lessons and practice exercises on function evaluation, composition, and much more. You can find them by searching for "Khan Academy algebra functions". Another excellent resource for understanding mathematical concepts deeply is Brilliant.org, which provides interactive problem-solving courses that can significantly enhance your grasp of these topics through engaging challenges. Exploring these resources will undoubtedly deepen your understanding and appreciation for the power and beauty of mathematical functions.