Combining Functions Exploring F(x) = X² And G(x) = -2x + 1

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Hey guys! Ever get that feeling when math problems look like a jumbled mess of symbols? Well, let's untangle some function fun today. We're going to break down two cool functions: f(x) = x² and g(x) = -2x + 1. We will explore how to combine them through addition and subtraction, and even visualize these new functions. So, grab your pencils (or styluses!), and let's dive in!

Understanding the Functions: f(x) = x² and g(x) = -2x + 1

Before we start mixing and matching these functions, let's get to know them a little better. These two functions are foundational in algebra, each with distinct characteristics and behaviors. Understanding these characteristics is key to grasping how they interact when combined.

Unpacking f(x) = x²: The Parabola

Let's kick things off with f(x) = x². This is a classic quadratic function, and its graph is a U-shaped curve called a parabola. The key feature of this function is that it squares any input value of x. This squaring action has some interesting consequences. For starters, whether x is positive or negative, f(x) will always be non-negative (zero or positive). Think about it: a positive number squared is positive, and a negative number squared is also positive. Zero squared is zero, of course. This means the parabola opens upwards and its lowest point, called the vertex, is at the origin (0, 0).

Another thing to notice is the symmetry of the parabola. Because squaring both x and -x gives the same result, the graph is symmetrical about the y-axis. If you were to fold the graph along the y-axis, the two halves would perfectly overlap. This symmetry is a hallmark of even functions, and f(x) = x² is a prime example. The rate at which the function increases isn't constant; it accelerates as x moves away from zero. This acceleration is what gives the parabola its curved shape, becoming steeper as the absolute value of x increases. The vertex at (0, 0) serves as a critical point where the function transitions from decreasing to increasing. For x values less than 0, the function decreases, and for x values greater than 0, it increases. This behavior is typical of quadratic functions with a positive leading coefficient. Knowing these properties will help visualize and manipulate the function when combined with others.

Dissecting g(x) = -2x + 1: The Straight Line

Now, let's turn our attention to g(x) = -2x + 1. This is a linear function, meaning its graph is a straight line. The general form of a linear function is g(x) = mx + b, where m is the slope and b is the y-intercept. In our case, m = -2 and b = 1. The slope of -2 tells us that the line slopes downwards as we move from left to right. For every 1 unit we move to the right along the x-axis, the line drops 2 units along the y-axis. This negative slope indicates a decreasing function; as x increases, g(x) decreases. The y-intercept of 1 tells us where the line crosses the y-axis. It's the point where x = 0, and in this case, the line intersects the y-axis at the point (0, 1). This point is crucial for plotting the graph of the line. Unlike the parabola of f(x), the linear function g(x) changes at a constant rate. This constant rate of change is what gives the line its straight shape. Linear functions are fundamental because they represent direct relationships and constant rates, making them simpler to analyze and predict. Understanding the slope and y-intercept provides a clear picture of how the function behaves and where it is positioned on the coordinate plane. This understanding will be valuable when we combine this function with f(x).

Combining Functions: Addition (f + g)(x)

Alright, now for the fun part: combining our functions! First up, we'll tackle addition. Adding functions is like adding their 'ingredients' together. We find (f + g)(x) by simply adding the expressions for f(x) and g(x). This process is straightforward but leads to a new function with combined characteristics of the original functions. Let's see how it works.

The Mechanics of Function Addition

So, to find (f + g)(x), we literally add f(x) and g(x) together. Mathematically, it looks like this:

(f + g)(x) = f(x) + g(x)

We know that f(x) = x² and g(x) = -2x + 1. So, we substitute these expressions into our equation:

(f + g)(x) = x² + (-2x + 1)

Now, we simplify by removing the parentheses and combining like terms:

(f + g)(x) = x² - 2x + 1

And there you have it! The sum of our functions, (f + g)(x), is the new function x² - 2x + 1. This resulting function is also a quadratic function, like f(x), but it has been transformed by the addition of the linear function g(x). The coefficients have changed, which affects the shape and position of the parabola. Recognizing this is crucial for understanding the behavior of the combined function.

Analyzing the Resulting Function

Notice that (f + g)(x) = x² - 2x + 1 is still a quadratic function, but its graph will be a parabola that's been shifted compared to the original f(x) = x². This new parabola has the same U-shape, but its position in the coordinate plane is different. To understand how the graph has changed, we can look at its vertex and its symmetry. This parabola opens upwards because the coefficient of the x² term is positive (1). The vertex of this parabola can be found by completing the square or using the formula x = -b / 2a. In this case, a = 1 and b = -2, so the x-coordinate of the vertex is x = -(-2) / (2 * 1) = 1. Substituting x = 1 back into the function gives the y-coordinate of the vertex: (1)² - 2(1) + 1 = 0. Thus, the vertex is at the point (1, 0). This shift of the vertex from (0, 0) in f(x) to (1, 0) in (f + g)(x) is a key change resulting from the addition. Furthermore, the axis of symmetry for this parabola is the vertical line x = 1, which passes through the vertex. This line divides the parabola into two mirror-image halves. Understanding these transformations—the shift in vertex and the axis of symmetry—helps in accurately graphing and analyzing the function (f + g)(x). The combined function presents a modified version of the original quadratic, showing how adding a linear function affects the shape and position of a parabola.

Combining Functions: Subtraction (f - g)(x)

Now, let's switch gears and try subtraction! Subtracting functions is similar to addition, but we're now finding the difference between the function 'ingredients'. This operation is equally straightforward and yields a function that reflects the subtraction of g(x)'s characteristics from f(x). The order of subtraction matters here, as subtracting g(x) from f(x) is different from subtracting f(x) from g(x). Let's see how this works out.

The Mechanics of Function Subtraction

To find (f - g)(x), we subtract the expression for g(x) from the expression for f(x). Here's how it looks mathematically:

(f - g)(x) = f(x) - g(x)

We know f(x) = x² and g(x) = -2x + 1, so we plug these in:

(f - g)(x) = x² - (-2x + 1)

Now, we simplify. Remember, subtracting a negative is like adding a positive, and we need to distribute the negative sign across the terms in the parentheses:

(f - g)(x) = x² + 2x - 1

So, (f - g)(x) turns out to be x² + 2x - 1. Again, we have a quadratic function, but this time, the coefficients are different from both f(x) and (f + g)(x). This new set of coefficients affects the parabola's position and shape differently compared to the previous operations. It highlights how sensitive the resulting function is to the operation performed.

Analyzing the Resulting Function

Like (f + g)(x), the function (f - g)(x) = x² + 2x - 1 is also a quadratic function, so its graph is a parabola. This parabola opens upwards since the coefficient of x² is positive. However, subtracting g(x) from f(x) has resulted in a different shift and position compared to adding g(x). To find the vertex of this parabola, we can use the formula x = -b / 2a. Here, a = 1 and b = 2, so the x-coordinate of the vertex is x = -2 / (2 * 1) = -1. Now, we substitute x = -1 back into the function to find the y-coordinate of the vertex: (-1)² + 2(-1) - 1 = 1 - 2 - 1 = -2. Therefore, the vertex of the parabola is at the point (-1, -2). This significant shift from the origin in f(x) and the different position in (f + g)(x) showcases the impact of subtraction. The axis of symmetry for this parabola is the vertical line x = -1, passing through the vertex. Visualizing this parabola helps to see how the subtraction of a linear function has transformed the original quadratic function. The change in the vertex position and the overall shape demonstrates the distinct effect of subtracting functions versus adding them, underlining the importance of the operation's order.

Visualizing Function Combinations: Graphing (f + g)(x) and (f - g)(x)

Okay, now that we've crunched the numbers, let's bring these functions to life with graphs! Graphing our combined functions (f + g)(x) and (f - g)(x) helps us visualize their behavior and see how the addition and subtraction of g(x) have transformed the original f(x). The visual representation provides a more intuitive understanding of the functions' characteristics, such as their vertices, axes of symmetry, and overall shape. This step is crucial for making connections between the algebraic manipulations and the geometric outcomes.

Graphing (f + g)(x) = x² - 2x + 1

We already figured out that (f + g)(x) = x² - 2x + 1 is a parabola, and we found its vertex to be at the point (1, 0). We also know it opens upwards since the coefficient of x² is positive. To sketch the graph, it's helpful to find a few more points. Let's calculate the values for a couple of x values on either side of the vertex. For example:

  • When x = 0, (f + g)(0) = (0)² - 2(0) + 1 = 1. So, we have the point (0, 1).
  • When x = 2, (f + g)(2) = (2)² - 2(2) + 1 = 1. So, we have the point (2, 1).

Plotting the vertex (1, 0) and these additional points (0, 1) and (2, 1), we can draw a smooth U-shaped curve that represents the graph of (f + g)(x). The graph is symmetrical about the vertical line x = 1, which is its axis of symmetry. This parabola is a transformation of the basic parabola f(x) = x², shifted one unit to the right. Visualizing this transformation helps to understand how adding a linear function to a quadratic function affects its graph. The new parabola retains the basic U-shape but is positioned differently in the coordinate plane, showing the impact of combining functions.

Graphing (f - g)(x) = x² + 2x - 1

Similarly, we know that (f - g)(x) = x² + 2x - 1 is also a parabola, and we found its vertex to be at the point (-1, -2). It also opens upwards. To sketch this graph, let's find a few more points:

  • When x = 0, (f - g)(0) = (0)² + 2(0) - 1 = -1. So, we have the point (0, -1).
  • When x = -2, (f - g)(-2) = (-2)² + 2(-2) - 1 = 4 - 4 - 1 = -1. So, we have the point (-2, -1).

Plotting the vertex (-1, -2) and these additional points (0, -1) and (-2, -1), we can sketch the U-shaped curve for (f - g)(x). This parabola is symmetrical about the vertical line x = -1, its axis of symmetry. Compared to f(x) = x², this parabola has been shifted one unit to the left and two units down. The visual comparison between the graphs of f(x), (f + g)(x), and (f - g)(x) clearly illustrates the effects of function addition and subtraction. Each operation results in a distinct transformation of the original parabola, demonstrating the versatility and interconnectedness of algebraic and geometric representations. Graphing these functions solidifies the understanding of how function combinations change their graphical properties.

Wrapping Up

So, there you have it! We've taken a good look at the functions f(x) = x² and g(x) = -2x + 1, combined them using addition and subtraction, and even graphed the results. By understanding the characteristics of individual functions and how they interact when combined, you guys can tackle more complex function problems with confidence. Remember, math isn't just about formulas; it's about understanding the relationships and patterns that those formulas represent. Keep exploring, and happy function-ing!