Solving F(x) ≥ 0 Graphically: A Step-by-Step Guide

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Solving f(x) ≥ 0 Graphically: A Step-by-Step Guide

Hey guys! Today, we're diving into a super practical problem: solving inequalities using graphs. Specifically, we'll tackle the inequality f(x) ≥ 0, where f(x) = (x+1)(x-3)^2. This might seem intimidating at first, but trust me, by the end of this guide, you'll be able to solve these types of problems like a pro. We’ll break it down step-by-step, making sure everyone can follow along. So, grab your pencils and let’s get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the problem is asking. We have a function, f(x) = (x+1)(x-3)^2, and we want to find all the values of x for which this function is greater than or equal to zero. In other words, we're looking for the intervals on the x-axis where the graph of f(x) is on or above the x-axis. This is where the visual power of a graph comes in handy. A graph gives us a clear picture of the function's behavior, making it much easier to identify these intervals. The key here is to understand the relationship between the function's values and its graphical representation. When f(x) is positive, the graph is above the x-axis; when f(x) is negative, the graph is below the x-axis; and when f(x) = 0, the graph intersects the x-axis. This visual approach not only helps in solving inequalities but also deepens our understanding of functions and their properties. So, let’s use this visual approach to crack this problem! We will go through each step carefully, ensuring that you grasp the underlying concepts and can apply them to similar problems in the future.

Step 1: Find the Zeros of the Function

First things first, let's find the zeros of the function. The zeros are the x-values where f(x) = 0. These are the points where the graph intersects the x-axis, and they are crucial for determining the intervals where the function is positive or negative. To find the zeros, we set f(x) = 0 and solve for x:

(x+1)(x-3)^2 = 0

This equation is already factored, which is super convenient! We can see that the product of two factors is zero if either factor is zero. So, we have two possibilities:

  1. x + 1 = 0 This gives us x = -1.
  2. (x - 3)^2 = 0 This gives us x - 3 = 0, which means x = 3.

So, the zeros of the function are x = -1 and x = 3. These are the x-coordinates where the graph of f(x) touches or crosses the x-axis. Notice that x = 3 is a repeated root because the factor (x - 3) is squared. This means that the graph will touch the x-axis at x = 3 but won't cross it. The behavior of the graph at repeated roots is a key concept in understanding the function's overall shape and how it relates to the inequality we're trying to solve. We'll see this more clearly in the next steps when we analyze the graph's intervals.

Step 2: Sketch the Graph of the Function

Now that we have the zeros, let's sketch the graph of the function. We don't need a perfectly accurate graph, just a rough sketch that shows the key features. To do this effectively, consider the following points:

  • Zeros: We know the graph intersects the x-axis at x = -1 and x = 3. Plot these points.
  • Behavior at Zeros: At x = -1, the factor (x + 1) changes sign, so the graph will cross the x-axis. At x = 3, the factor (x - 3)^2 does not change sign (it's always non-negative), so the graph will touch the x-axis and bounce back.
  • End Behavior: For large positive x, f(x) will be positive because all factors will be positive. For large negative x, f(x) will be positive because (x - 3)^2 is always positive and (x + 1) will be negative, making the product positive.

With this information, we can sketch a curve that:

  • Crosses the x-axis at x = -1.
  • Touches the x-axis at x = 3 and bounces back.
  • Is positive for large positive and negative x.

The sketch should look like a curve that comes from the top-left, crosses the x-axis at x = -1, goes down and then comes back up to touch the x-axis at x = 3, and then goes up again. Remember, this is a sketch, so the exact shape isn't crucial, but the key features (zeros and behavior at zeros) are important. A good sketch helps visualize the intervals where the function is positive, negative, or zero, which is the key to solving the inequality.

Step 3: Identify Intervals Where f(x) ≥ 0

Now, the fun part: identifying the intervals where f(x) ≥ 0. This means we're looking for the sections of the x-axis where the graph is on or above the x-axis. Looking at our sketch, we can see that this occurs in two main regions:

  1. To the right of x = -1: The graph is above the x-axis for all x values greater than -1. This interval includes the point x = 3, where the graph touches the x-axis.
  2. At x = 3: Even though the graph touches the x-axis at x = 3, this point is included in our solution because we are looking for f(x) ≥ 0, which includes the case where f(x) = 0.

Therefore, the solution to the inequality f(x) ≥ 0 is all x in the interval [-1, ∞). This means that for any x value greater than or equal to -1, the function f(x) will be greater than or equal to zero. The interval notation helps us concisely express the solution set. Remember, the square bracket indicates that the endpoint is included in the solution (because of the "equal to" part of the inequality), and the infinity symbol indicates that the interval extends indefinitely in the positive direction.

Step 4: Write the Solution in Interval Notation

Finally, let's write our solution in interval notation. This is a standard way to express sets of numbers, and it's super useful for communicating mathematical ideas clearly. As we determined in the previous step, the solution to the inequality f(x) ≥ 0 is all x values greater than or equal to -1. In interval notation, this is written as:

[-1, ∞)

Let's break this down:

  • The square bracket '[' indicates that -1 is included in the solution because f(-1) = 0, and we want f(x) ≥ 0.
  • The parenthesis ')' indicates that infinity is not included in the solution (because infinity is not a number, but a concept representing unboundedness).
  • The interval [-1, ∞) means all real numbers from -1 up to infinity.

So, our final answer, expressed in interval notation, is [-1, ∞). This notation provides a concise and precise way to represent the solution set, making it easy to communicate the result to others. Mastering interval notation is a valuable skill in mathematics, as it's used extensively in various areas, including calculus and analysis.

Conclusion

And there you have it! We've successfully solved the inequality f(x) ≥ 0, where f(x) = (x+1)(x-3)^2, by using the graph of the function. We broke it down into easy-to-follow steps:

  1. Finding the zeros of the function.
  2. Sketching the graph.
  3. Identifying the intervals where f(x) ≥ 0.
  4. Writing the solution in interval notation.

Remember, the key to solving inequalities graphically is to visualize the function's behavior. The graph tells you everything you need to know about where the function is positive, negative, or zero. This method isn't just useful for this specific problem; it's a powerful tool that can be applied to a wide range of inequalities. By understanding the relationship between a function and its graph, you can tackle these problems with confidence and ease. So, keep practicing, and you'll become a pro at solving inequalities graphically in no time! Keep up the great work, guys!