Find F(-6) For F(x) = -4 - 2x

When we're diving into the world of functions in mathematics, one of the most fundamental concepts is evaluating a function at a specific value. The function we're looking at today is $f(x) = -4 - 2x$, and our mission is to find the value of $f(-6)$. This process is straightforward and forms the bedrock for understanding more complex mathematical operations. Think of a function like a machine: you put something in (the input), and it gives you something out (the output). In this case, the function $f$ takes an input $x$, performs the operation $-4 - 2x$, and returns the result. Our goal is to see what happens when we input the number -6 into this particular machine. It's like asking, "What's the output if the input is -6?"

Understanding Function Notation

Before we jump into the calculation, let's quickly recap what the notation $f(x) = -4 - 2x$ means. The 'f' represents the name of the function. The '(x)' inside the parentheses indicates that the function's output depends on the variable 'x'. So, $f(x)$ is essentially a placeholder for the output value of the function when the input is 'x'. The equation $f(x) = -4 - 2x$ tells us precisely how to calculate that output: take the input 'x', multiply it by -2, and then subtract 4 from the result. So, if we wanted to find $f(3)$, we would replace every 'x' in the expression $-4 - 2x$ with the number 3 and compute the result: $f(3) = -4 - 2(3) = -4 - 6 = -10$. It's a systematic way to map inputs to outputs. The notation is incredibly powerful because it allows us to define relationships between variables concisely and unambiguously. When we see $f(-6)$, it's a clear instruction to substitute $-6$ for every occurrence of $x$ in the function's definition.

The Substitution Process

Now, let's get to the heart of the matter: calculating $f(-6)$ for the function $f(x) = -4 - 2x$. The notation $f(-6)$ explicitly tells us to substitute $-6$ for every instance of $x$ in the expression $-4 - 2x$. So, we rewrite the function's definition, replacing $x$ with $(-6)$: $f(-6) = -4 - 2(-6)$. The parentheses around $-6$ are crucial here, especially when dealing with negative numbers and multiplication, to avoid any confusion. This step is where the magic of function evaluation truly happens – we're applying the function's rule to a specific number. It's a direct application of the function's definition, transforming the abstract expression into a concrete numerical value. This substitution is the key step that allows us to move from a general rule ($f(x)$) to a specific outcome ($f(-6)$). Once the substitution is made, the problem transforms into a simple arithmetic calculation, testing our understanding of order of operations and operations with negative numbers.

Performing the Arithmetic

With the substitution made, we now have the expression $f(-6) = -4 - 2(-6)$. The next step is to carefully perform the arithmetic operations according to the standard order of operations (PEMDAS/BODMAS). First, we handle the multiplication: $-2$ multiplied by $-6$. Remember that multiplying two negative numbers results in a positive number. So, $-2 imes -6 = 12$. Now, we substitute this result back into our expression: $f(-6) = -4 + 12$. The final step is a simple addition. We are adding $-4$ and $12$. This is equivalent to $12 - 4$. Performing this subtraction gives us $8$. Therefore, the value of $f(-6)$ is $8$. This calculation demonstrates the importance of correctly handling signs when working with negative numbers. A common mistake could be to incorrectly multiply $-2$ by $-6$, perhaps getting $-12$ instead of $12$, which would lead to an incorrect final answer. But by carefully following the rules of arithmetic, we arrive at the definitive value. This is a clear illustration of how algebraic expressions translate into numerical results through precise calculation.

Conclusion: The Value of f(-6)

In summary, by taking the function $f(x) = -4 - 2x$ and substituting $-6$ for $x$, we performed the necessary arithmetic operations. The multiplication of $-2$ by $-6$ yielded $12$, and then adding this to $-4$ resulted in $8$. Thus, the value of $f(-6)$ is $8$. This exercise highlights the fundamental concept of function evaluation, a skill that is essential for many areas of mathematics, from algebra to calculus and beyond. Understanding how to input values into functions and interpret the outputs is a critical step in building a strong mathematical foundation. It allows us to explore relationships between numbers and understand how changes in input affect outcomes. For further exploration into functions and their properties, you can visit ** **Khan Academy.