Evaluating Functions A Step By Step Guide To Solving F(3) For F(x) = 2x² + 1
In the realm of mathematics, functions play a crucial role in describing relationships between variables. Evaluating a function at a specific point is a fundamental skill, allowing us to determine the output of the function for a given input. This article delves into the process of evaluating a function, using the example of f(x) = 2x² + 1. We will explore how to find the value of f(3), providing a step-by-step guide and elucidating the underlying concepts. Understanding function evaluation is paramount for various mathematical applications, from solving equations to graphing functions and modeling real-world phenomena.
Demystifying Function Notation and Evaluation
Before we dive into the specific example, let's clarify the notation and the general process of function evaluation. A function, often denoted by f(x), represents a rule that assigns a unique output value for each input value x. The expression f(x) is read as "f of x" and signifies the value of the function f at the input x. The variable x is called the independent variable, and the output value f(x) is called the dependent variable because its value depends on the value of x. Function evaluation involves substituting a specific value for the independent variable x in the function's expression and simplifying the result. This process yields the corresponding output value of the function at that particular input.
To evaluate a function, we replace every instance of the input variable in the function's formula with the given value. For instance, if we have a function g(x) = x + 5, to evaluate g(2), we would substitute 2 for x in the expression, obtaining g(2) = 2 + 5 = 7. This means that when the input to the function g is 2, the output is 7. The same principle applies to more complex functions involving powers, radicals, or other mathematical operations. Careful substitution and adherence to the order of operations are essential to arrive at the correct result. This meticulous approach ensures accuracy and avoids common pitfalls in function evaluation.
Step-by-Step Solution: Evaluating f(3) for f(x) = 2x² + 1
Now, let's tackle the problem at hand: finding f(3) when f(x) = 2x² + 1. This function is a quadratic function, which means it involves a term with x raised to the power of 2. The expression 2x² + 1 indicates that for any input x, we first square x, then multiply the result by 2, and finally add 1. To evaluate f(3), we will follow these steps:
- Substitute 3 for x in the function's expression: This gives us f(3) = 2(3)² + 1. We are replacing the variable x with the numerical value 3 to determine the function's output at this specific input. The parentheses around 3 in (3)² emphasize that the entire value 3 is being squared.
- Evaluate the exponent: According to the order of operations (PEMDAS/BODMAS), we perform exponentiation before multiplication and addition. So, we calculate 3², which is 3 multiplied by itself: 3² = 3 * 3 = 9. Our expression now becomes f(3) = 2(9) + 1.
- Perform the multiplication: Next, we multiply 2 by 9: 2 * 9 = 18. This simplifies our expression to f(3) = 18 + 1. The multiplication step is crucial in determining the coefficient of the quadratic term's contribution to the final result.
- Perform the addition: Finally, we add 18 and 1: 18 + 1 = 19. Therefore, f(3) = 19. This final addition completes the calculation and provides the output value of the function at the input x = 3. We have successfully evaluated the function f(x) at the specified point.
Thus, we have determined that when x is 3, the value of the function f(x) = 2x² + 1 is 19. This means the point (3, 19) lies on the graph of this quadratic function. The process we followed illustrates the general method for evaluating any function at a given point, regardless of its complexity. By substituting the input value for the variable and simplifying the expression according to the order of operations, we can find the corresponding output value.
Visualizing Function Evaluation: Graphing the Function
To further understand the concept of function evaluation, it's helpful to visualize the function's graph. The graph of f(x) = 2x² + 1 is a parabola, a U-shaped curve. Each point on the graph represents a pair of input and output values (x, f(x)). The point (3, 19), which we found by evaluating f(3), lies on this parabola. We can plot other points by evaluating the function at different values of x, such as f(0) = 1, f(1) = 3, f(-1) = 3, and so on.
The graphical representation provides a visual confirmation of our algebraic calculation. By locating the point on the graph where the x-coordinate is 3, we can see that the corresponding y-coordinate (the function value) is indeed 19. This visual connection between the algebraic expression and the geometric representation reinforces the understanding of function evaluation. Moreover, graphing the function allows us to observe its behavior and identify key features, such as the vertex, axis of symmetry, and intercepts.
The Significance of Function Evaluation in Mathematics
Function evaluation is a cornerstone of mathematical analysis and has widespread applications in various fields. It enables us to analyze the behavior of functions, solve equations, and model real-world phenomena. In calculus, function evaluation is crucial for finding limits, derivatives, and integrals. In algebra, it is used to solve systems of equations and analyze polynomial functions. In physics and engineering, functions are used to describe motion, forces, and other physical quantities, and evaluating these functions at specific times or locations allows us to make predictions and solve problems.
Understanding how to evaluate functions is also essential for interpreting data and making informed decisions. For example, if we have a function that models the growth of a population over time, evaluating the function at a particular time will give us an estimate of the population size at that time. Similarly, if we have a function that represents the cost of producing a certain number of items, evaluating the function will tell us the cost of producing a specific quantity. The ability to evaluate functions empowers us to make quantitative assessments and gain insights from mathematical models.
Practice Problems: Strengthening Your Understanding
To solidify your understanding of function evaluation, let's consider a few practice problems:
- If g(x) = x³ - 2x + 5, find g(-2).
- If h(t) = √(t + 4), find h(5).
- If p(r) = (3r² - 1) / (r + 2), find p(0).
Solving these problems will reinforce the steps involved in function evaluation and help you become more confident in applying the concept. Remember to substitute the given input value for the variable, follow the order of operations, and simplify the expression carefully. Checking your answers and reviewing the process will further enhance your understanding.
Conclusion: Mastering Function Evaluation
In conclusion, evaluating a function at a specific point is a fundamental mathematical skill with broad applications. By substituting the input value for the variable and simplifying the expression, we can determine the corresponding output value. The example of f(x) = 2x² + 1 and the evaluation of f(3) demonstrated the step-by-step process involved. Visualizing the function's graph provides a geometric interpretation of function evaluation, further enhancing our understanding. Function evaluation is essential for analyzing the behavior of functions, solving equations, and modeling real-world phenomena. Mastering this skill is crucial for success in mathematics and related fields. By practicing and applying the concepts discussed in this article, you can develop a strong foundation in function evaluation and unlock its power in various mathematical contexts.