Evaluate E(-4) Given E(x) = X^2 - X + 20

Let's dive into evaluating a function! In this article, we're going to tackle the problem of finding the value of e(-4), given that e(x) = x^2 - x + 20. This is a common type of problem in algebra and calculus, and it's all about understanding how functions work. We will carefully substitute -4 into the function e(x) and follow the order of operations to get to the final answer. So, let's get started!

Understanding the Function

Before we jump into the calculation, it's essential to understand what the function e(x) = x^2 - x + 20 represents. A function is like a machine that takes an input, does something to it, and spits out an output. In this case, our function e(x) takes a number x, squares it, subtracts the original number x, and then adds 20. Think of it as a recipe: you give it an ingredient (x), and it follows a set of instructions to produce a dish (e(x)).

In mathematical terms, e(x) is a quadratic function. Quadratic functions are characterized by having a term with x raised to the power of 2 (x^2). The graph of a quadratic function is a parabola, a U-shaped curve. Understanding the nature of the function can help us predict its behavior and understand the significance of the values we calculate.

For example, the x^2 term means that the function will grow rapidly as x gets larger in either the positive or negative direction. The -x term introduces a slight asymmetry, shifting the parabola slightly. And the +20 simply moves the entire graph upwards by 20 units. Remember, e(x) is just notation, it could be any letter really such as f(x) or g(x). There are also other types of functions that behave differently, such as trigonometric, logarithmic, and exponential functions. These functions have their own unique properties and behaviors, which are explored in more advanced mathematics courses.

Substituting -4 into the Function

Now that we understand the function, let's find e(-4). This means we need to replace every instance of x in the function's formula with -4. So, we have:

e(-4) = (-4)^2 - (-4) + 20

It's crucial to pay close attention to the signs when substituting negative numbers. Make sure to enclose the negative number in parentheses to avoid errors. For example, (-4)^2 is different from -4^2. In the first case, we're squaring the entire number -4, while in the second case, we're squaring 4 and then negating the result. Using parentheses ensures that we perform the operations in the correct order.

Substituting values into functions is a fundamental operation in mathematics. It allows us to explore the behavior of functions at specific points and to understand how the output of a function changes as we vary the input. This process is used extensively in calculus, where we study the rates of change of functions and their properties.

Evaluating the Expression

Now that we've substituted -4 into the function, we need to evaluate the resulting expression. Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Following the order of operations, we first evaluate the exponent:

(-4)^2 = (-4) * (-4) = 16

Remember that a negative number multiplied by a negative number gives a positive number. Next, we handle the subtraction of a negative number:

-(-4) = +4

Subtracting a negative number is the same as adding its positive counterpart. Now, we can rewrite the expression as:

e(-4) = 16 + 4 + 20

Finally, we perform the addition:

e(-4) = 16 + 4 + 20 = 40

So, e(-4) = 40. This means that when we input -4 into the function e(x), the output is 40.

Conclusion

We have successfully evaluated e(-4) given the function e(x) = x^2 - x + 20. By understanding the function, substituting the value correctly, and following the order of operations, we arrived at the answer: e(-4) = 40. This exercise demonstrates the fundamental principles of working with functions and evaluating expressions, which are essential skills in mathematics and related fields.

Understanding functions and how to evaluate them is crucial for many applications in science and engineering. Whether you're calculating the trajectory of a rocket, modeling the spread of a disease, or designing a new algorithm, functions are the building blocks of mathematical models.

Keep practicing with different functions and values to build your confidence and understanding. The more you work with functions, the more comfortable you'll become with their properties and behaviors.

For further learning and practice, you might find it helpful to explore resources on function evaluation and algebra. A great resource would be Khan Academy's Algebra I course. Keep up the great work, and happy calculating!