Solving And Evaluating The Function F(x) = 2x + 6

by Admin 50 views

This article delves into the function f(x) = 2x + 6, exploring its evaluation at a specific point and solving for its roots. We will break down each step in detail, providing a comprehensive understanding of the underlying mathematical concepts.

a) Evaluating f(-4)

In this section, we focus on evaluating the function f(x) = 2x + 6 at x = -4. This process involves substituting -4 for x in the function's equation and simplifying the expression. Understanding function evaluation is crucial in mathematics as it allows us to determine the output of a function for a given input. This concept is fundamental in various applications, including graphing functions, modeling real-world scenarios, and solving equations. Function evaluation lays the groundwork for more advanced mathematical concepts, such as calculus and differential equations. Let's dive into the step-by-step process of evaluating f(-4).

To evaluate f(-4), we begin by replacing every instance of 'x' in the function's equation with '-4'. The function f(x) = 2x + 6 then transforms into f(-4) = 2(-4) + 6. The next step involves performing the multiplication operation. We multiply 2 by -4, which gives us -8. This changes our equation to f(-4) = -8 + 6. Finally, we perform the addition. Adding -8 and 6 results in -2. Therefore, f(-4) = -2. This result signifies that when the input of the function is -4, the corresponding output is -2. Graphically, this represents the point (-4, -2) on the graph of the function. This process of substituting a value into a function and simplifying is the core of function evaluation. It allows us to map inputs to outputs, providing a complete understanding of the function's behavior.

The initial evaluation provided f(-4) = -2, which is the correct answer. The process of substituting -4 into the function and simplifying the expression demonstrates a clear understanding of function evaluation. This fundamental skill is essential for more advanced mathematical concepts and applications. By correctly evaluating functions, we can accurately predict outputs for given inputs, which is crucial in various fields such as physics, engineering, and computer science. The ability to evaluate functions efficiently also aids in solving equations and inequalities, allowing us to determine the values of variables that satisfy specific conditions. In summary, mastering function evaluation is a cornerstone of mathematical proficiency.

b) Solving f(x) = 0

Next, let's tackle the task of solving the equation f(x) = 0 for the given function f(x) = 2x + 6. Solving an equation means finding the value(s) of the variable (in this case, 'x') that make the equation true. In the context of functions, solving f(x) = 0 is equivalent to finding the x-intercept(s) of the function's graph, which are the points where the graph crosses the x-axis. These points are also known as the roots or zeros of the function. Understanding how to solve equations is a fundamental skill in algebra and is essential for various mathematical applications, including finding the solutions to real-world problems modeled by equations. Let's break down the steps involved in solving f(x) = 0.

To solve f(x) = 0, we first substitute the function's expression, 2x + 6, for f(x). This gives us the equation 2x + 6 = 0. Our goal is to isolate 'x' on one side of the equation. The first step in achieving this is to subtract 6 from both sides of the equation. This maintains the balance of the equation and eliminates the constant term on the left side. Subtracting 6 from both sides gives us 2x = -6. Now, to completely isolate 'x', we need to divide both sides of the equation by the coefficient of 'x', which is 2. Dividing both sides by 2 gives us x = -3. This solution, x = -3, represents the value of 'x' that makes the function f(x) equal to zero. Graphically, this means that the graph of the function f(x) = 2x + 6 intersects the x-axis at the point (-3, 0).

Therefore, the solution to the equation f(x) = 0 is x = -3. This demonstrates the process of solving a linear equation, a core skill in algebra. Solving equations allows us to find specific values that satisfy certain conditions, which is crucial in various mathematical and scientific applications. The ability to manipulate equations and isolate variables is essential for problem-solving in a wide range of fields. The solution x = -3 represents the root of the function, the point where the function's output is zero. This concept is fundamental in understanding the behavior of functions and their graphs. By mastering equation-solving techniques, we can effectively analyze and solve problems that involve mathematical relationships.

In conclusion, we have successfully evaluated the function f(x) = 2x + 6 at x = -4 and solved the equation f(x) = 0. These exercises demonstrate fundamental concepts in algebra and function analysis, skills essential for further mathematical studies and applications.