Evaluating Functions Finding F(2) For F(x) = X² + 4x + 3
Understanding Function Evaluation
In mathematics, a function is a fundamental concept that describes a relationship between an input and an output. It's like a machine: you feed it something (the input), and it produces something else (the output) based on a specific rule. This rule is defined by the function's equation. To evaluate a function, we substitute a given value for the variable (usually 'x') in the function's equation and then simplify the expression to find the corresponding output. This process allows us to understand the function's behavior at different points and is crucial for various mathematical applications, including graphing functions, solving equations, and modeling real-world phenomena.
Think of a function as a recipe. The function's equation is the list of ingredients and instructions, the input is the specific amount of each ingredient you want to use, and the output is the final dish you create. Just as different amounts of ingredients will result in different dishes, different inputs to a function will produce different outputs. Evaluating a function is like following the recipe with a specific set of ingredients to see what dish you'll get. Understanding function evaluation is essential for grasping more advanced mathematical concepts, such as calculus and differential equations, where the behavior of functions is analyzed in detail. Function evaluation also finds practical applications in various fields, including physics, engineering, economics, and computer science, where mathematical models are used to represent and analyze real-world systems. For example, a function might represent the trajectory of a projectile, the growth of a population, or the cost of production. By evaluating the function at different points, we can gain insights into the behavior of these systems and make predictions about their future states. In essence, function evaluation provides a powerful tool for understanding and working with mathematical relationships.
Problem Statement: Evaluating f(x) = x² + 4x + 3 at x = 2
Our task is to evaluate the function f(x) = x² + 4x + 3 at x = 2. This means we need to find the value of the function when we substitute '2' for 'x' in the equation. In simpler terms, we're asking: what is the output of the function when the input is 2? This is a classic example of function evaluation, and it demonstrates a core concept in algebra and calculus. The function f(x) = x² + 4x + 3 is a quadratic function, meaning it has a squared term (x²) as its highest power. Quadratic functions are particularly important in mathematics and have a wide range of applications. They describe parabolic curves, which appear in various physical phenomena, such as the trajectory of a ball thrown in the air or the shape of a satellite dish. Understanding how to evaluate quadratic functions is therefore crucial for understanding these phenomena. Evaluating a function at a specific point like x = 2 gives us a single, numerical value that represents the function's height at that point on its graph. This point is a specific location on the parabola defined by the quadratic function. By evaluating the function at multiple points, we can build a more complete picture of the parabola's shape and behavior. This process of function evaluation is not limited to quadratic functions; it applies to all types of functions, including linear, exponential, trigonometric, and logarithmic functions. Each type of function has its own unique characteristics and applications, but the basic principle of evaluation remains the same: substitute the input value for the variable in the function's equation and simplify to find the output value. The ability to evaluate functions is a fundamental skill in mathematics that allows us to connect abstract equations to concrete numerical values and to visualize the relationships between inputs and outputs. It forms the basis for many more advanced mathematical concepts and techniques.
Step-by-Step Solution
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Substitute x = 2 into the function:
We begin by replacing every instance of 'x' in the function's equation with the value '2'. This gives us:
f(2) = (2)² + 4(2) + 3
This substitution is the heart of function evaluation. We are essentially asking the function to perform its operations on the specific input value of 2. The parentheses are crucial here, as they ensure that the substitution is done correctly and that the order of operations is followed accurately. The substitution transforms the function's equation from a general expression involving the variable 'x' into a specific numerical expression that we can now simplify. The result of this substitution represents the value of the function at the point x = 2. It's important to note that this substitution process is not unique to this particular function; it's a general method that applies to all functions, regardless of their complexity. Whether the function is a simple linear equation or a complex trigonometric function, the first step in evaluating it at a specific point is always to substitute that value for the variable in the function's equation. This initial step sets the stage for the subsequent simplification steps, which will ultimately lead us to the function's output value at the given input.
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Simplify the expression using the order of operations (PEMDAS/BODMAS):
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Exponents: First, we evaluate the exponent: (2)² = 4
Now our expression looks like:
f(2) = 4 + 4(2) + 3
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Multiplication: Next, we perform the multiplication: 4(2) = 8
Our expression becomes:
f(2) = 4 + 8 + 3
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Addition: Finally, we add the remaining terms:
f(2) = 15
The order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), is a fundamental principle in mathematics that ensures consistency and accuracy in calculations. It dictates the sequence in which mathematical operations should be performed to arrive at the correct answer. In this case, following the order of operations is crucial for correctly evaluating the function. First, we address the exponent, squaring the value 2. This operation takes precedence over multiplication and addition. Next, we perform the multiplication, multiplying 4 by 2. Finally, we complete the calculation by adding the remaining terms: 4, 8, and 3. Adhering to the order of operations is not just a matter of convention; it's essential for obtaining the correct result. If we were to perform the addition before the multiplication, for example, we would arrive at a different and incorrect answer. The order of operations ensures that mathematical expressions are interpreted and evaluated unambiguously, preventing errors and maintaining the integrity of mathematical calculations. Mastering the order of operations is a cornerstone of mathematical proficiency, and it's a skill that is essential for success in algebra, calculus, and other advanced mathematical topics. It also has practical applications in various fields, such as engineering, physics, and computer science, where accurate calculations are critical.
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Final Answer
Therefore, f(2) = 15. This means that the value of the function f(x) = x² + 4x + 3 when x is equal to 2 is 15. This single numerical value provides a specific point on the graph of the function, indicating its height at x = 2. In the context of a quadratic function, this point lies on the parabola that represents the function's overall shape. The value f(2) = 15 is not just an isolated result; it's a piece of a larger picture. By evaluating the function at other values of x, we can obtain additional points and begin to trace out the entire parabola. This process of evaluating the function at multiple points is fundamental to understanding its behavior and its graphical representation. The concept of function evaluation extends beyond simple numerical calculations. It provides a bridge between the abstract equation of a function and its concrete behavior. The value of the function at a particular point tells us something about the function's overall characteristics and its relationship to the input variable. In this case, f(2) = 15 gives us a specific data point that can be used to analyze the quadratic function f(x) = x² + 4x + 3. This analysis might involve finding the function's minimum or maximum value, its intercepts with the x-axis, or its symmetry. Ultimately, function evaluation is a powerful tool for exploring and understanding mathematical relationships.
In conclusion, we have successfully evaluated the function f(x) = x² + 4x + 3 at x = 2 by substituting the value and simplifying the expression using the order of operations. This process demonstrates the fundamental concept of function evaluation, which is crucial for understanding and working with mathematical functions in various contexts.
