Solving For X When G(x) Equals -62 Given G(x) = 8x + 2
Let's dive into the fascinating world of functions and explore the linear function g(x) = 8x + 2. This function takes an input value, x, multiplies it by 8, and then adds 2 to the result. Understanding how functions work is a fundamental concept in mathematics, opening doors to more advanced topics like calculus and analysis. In this article, we will dissect this function, learn how to evaluate it for specific inputs, and tackle the challenge of finding the input value that produces a given output, specifically g(x) = -62.
Understanding the Function g(x) = 8x + 2
At its core, the function g(x) = 8x + 2 is a linear function. Linear functions are characterized by their straight-line graphs and can be represented in the general form of f(x) = mx + b, where 'm' represents the slope and 'b' represents the y-intercept. In our case, g(x) = 8x + 2, the slope (m) is 8, indicating the steepness of the line, and the y-intercept (b) is 2, representing the point where the line crosses the vertical y-axis. The slope of 8 means that for every increase of 1 in the x-value, the g(x) value increases by 8. The y-intercept of 2 tells us that when x is 0, g(x) is 2. To further illustrate this, let's consider a few examples. If we input x = 0 into the function, we get g(0) = 8(0) + 2 = 2. If we input x = 1, we get g(1) = 8(1) + 2 = 10. Notice how the output value increases by 8 as the input value increases by 1, confirming our understanding of the slope. This consistent rate of change is a defining characteristic of linear functions.
Graphically, this function would appear as a straight line on a coordinate plane. The line would rise sharply from left to right due to the positive slope of 8. It would intersect the y-axis at the point (0, 2), which is the y-intercept. Visualizing the graph can provide an intuitive understanding of how the function behaves. For instance, we can imagine moving along the line and observe how the g(x) value changes as we move along the x-axis. This graphical representation reinforces the concept of a linear relationship between x and g(x).
Evaluating g(x) for Different Inputs
Evaluating a function means finding the output value (g(x)) for a specific input value (x). To evaluate g(x) = 8x + 2 for a given value of x, we simply substitute that value into the function and perform the arithmetic operations. This process is crucial for understanding the function's behavior and predicting its output for various inputs. For example, let's evaluate g(x) for x = 3. We substitute x = 3 into the function: g(3) = 8(3) + 2. Following the order of operations, we first multiply 8 by 3, which gives us 24. Then, we add 2 to the result, giving us g(3) = 24 + 2 = 26. Therefore, when x is 3, g(x) is 26. We can repeat this process for any value of x.
Let's consider another example, this time with a negative input. Let's evaluate g(x) for x = -1. Substituting x = -1 into the function, we get g(-1) = 8(-1) + 2. Multiplying 8 by -1 gives us -8. Adding 2 to -8 gives us g(-1) = -8 + 2 = -6. Thus, when x is -1, g(x) is -6. This demonstrates that we can evaluate the function for both positive and negative input values. Furthermore, we can even evaluate the function for fractional or decimal values of x. For instance, if we want to find g(0.5), we substitute x = 0.5 into the function: g(0.5) = 8(0.5) + 2 = 4 + 2 = 6. This ability to evaluate the function for a wide range of input values allows us to gain a comprehensive understanding of its behavior.
Solving for x when g(x) = -62
Now, let's tackle the core question: finding the value of x when g(x) = -62. This involves working backward from the output to find the corresponding input. We are given the equation g(x) = -62, and we know that g(x) = 8x + 2. Therefore, we can set up the equation 8x + 2 = -62. To solve for x, we need to isolate it on one side of the equation. The first step is to subtract 2 from both sides of the equation. This maintains the equality and moves us closer to isolating x. Subtracting 2 from both sides, we get 8x + 2 - 2 = -62 - 2, which simplifies to 8x = -64. The next step is to divide both sides of the equation by 8. This will isolate x and give us the solution. Dividing both sides by 8, we get 8x / 8 = -64 / 8, which simplifies to x = -8. Therefore, the value of x that makes g(x) equal to -62 is x = -8. This means that when we input -8 into the function g(x) = 8x + 2, we get an output of -62.
To verify our solution, we can substitute x = -8 back into the original function and check if it yields -62. Substituting x = -8 into g(x) = 8x + 2, we get g(-8) = 8(-8) + 2. Multiplying 8 by -8 gives us -64. Adding 2 to -64 gives us g(-8) = -64 + 2 = -62. This confirms that our solution, x = -8, is correct. This process of solving for x when given g(x) is a fundamental skill in algebra and is used extensively in various mathematical applications. Understanding how to manipulate equations and isolate variables is crucial for solving a wide range of problems.
Stepping Through the Solution Process
Let's break down the process of solving for x when g(x) = -62 into clear, concise steps. This step-by-step approach will help solidify your understanding of the process and enable you to tackle similar problems with confidence.
Step 1: Set up the equation. We know that g(x) = 8x + 2 and we are given that g(x) = -62. Therefore, we set up the equation 8x + 2 = -62. This equation represents the relationship between the input x and the output -62 for the given function. It is the foundation for solving for the unknown value of x. Carefully setting up the equation is a crucial first step, as any errors at this stage will propagate through the rest of the solution.
Step 2: Isolate the term with x. To isolate the term with x, which is 8x, we need to eliminate the constant term, which is +2. We do this by performing the inverse operation, which is subtraction. We subtract 2 from both sides of the equation to maintain equality. This gives us 8x + 2 - 2 = -62 - 2, which simplifies to 8x = -64. By subtracting 2 from both sides, we have successfully isolated the term containing x on the left side of the equation.
Step 3: Solve for x. Now that we have 8x = -64, we need to isolate x completely. To do this, we divide both sides of the equation by the coefficient of x, which is 8. This is the inverse operation of multiplication. Dividing both sides by 8 gives us 8x / 8 = -64 / 8, which simplifies to x = -8. This is our solution: x = -8. We have successfully isolated x and found its value.
Step 4: Verify the solution (optional but recommended). To ensure that our solution is correct, we can substitute x = -8 back into the original function g(x) = 8x + 2 and see if it yields -62. Substituting x = -8, we get g(-8) = 8(-8) + 2 = -64 + 2 = -62. Since this matches the given output value, we can be confident that our solution x = -8 is correct. Verification is a crucial step in problem-solving, as it helps to catch any errors and ensures the accuracy of the solution. By following these steps, you can confidently solve for x in similar linear equations.
Conclusion
In this exploration, we've delved into the linear function g(x) = 8x + 2, understanding its components, evaluating it for various inputs, and, most importantly, solving for x when g(x) = -62. We discovered that x = -8 is the solution that satisfies the condition. This process highlights the fundamental concepts of function evaluation and equation solving, crucial skills in mathematics and beyond. The ability to understand and manipulate functions is essential for various applications, from modeling real-world phenomena to solving complex problems in science and engineering. By mastering these skills, you unlock a deeper understanding of the mathematical world and its applications.
