Finding The Roots Of F(x) = -3√(-2x - 3)

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Step 1: Understanding the Function

Before diving into the solution, let's first understand the function f(x) = -3√(-2x - 3). This is a radical function, specifically a square root function, multiplied by a constant. The key here is the square root, which imposes a restriction on the domain of the function. The expression inside the square root, also known as the radicand, must be non-negative for the function to produce real number outputs. This means -2x - 3 ≥ 0. Understanding this constraint is crucial for determining the validity of any potential roots we find.

The domain restriction is a fundamental concept in mathematics, particularly when dealing with radical functions and rational functions. It dictates the set of input values (x-values) for which the function is defined and produces real outputs. In the context of square root functions, the radicand (the expression under the square root) must be greater than or equal to zero. This is because the square root of a negative number is not a real number. Similarly, in rational functions (functions expressed as a fraction), the denominator cannot be zero, as division by zero is undefined.

In our specific case, the radicand is -2x - 3. To find the domain restriction, we set this expression greater than or equal to zero:

-2x - 3 ≥ 0

To solve this inequality, we follow these steps:

  1. Add 3 to both sides: -2x ≥ 3
  2. Divide both sides by -2. Remember, when dividing or multiplying an inequality by a negative number, we must flip the inequality sign: x ≤ -3/2

This inequality tells us that the function f(x) = -3√(-2x - 3) is only defined for x-values less than or equal to -3/2. This domain restriction is crucial because any potential root we find must fall within this interval to be considered a valid solution.

Step 2: Setting the Function to Zero

To find the roots, we set f(x) equal to zero:

-3√(-2x - 3) = 0

This equation represents the core of our problem. We need to find the x-value(s) that satisfy this equation. The left side of the equation is a product of -3 and a square root. For this product to be zero, one of the factors must be zero. Since -3 is a non-zero constant, the square root term must be zero:

√(-2x - 3) = 0

This simplification is a critical step in isolating x and solving for the root(s). We have now transformed the original equation into a more manageable form.

Step 3: Solving for x

To eliminate the square root, we square both sides of the equation:

(-2x - 3) = 0

Now we have a simple linear equation. To solve for x, we isolate the variable:

  1. Add 3 to both sides: -2x = 3
  2. Divide both sides by -2: x = -3/2

This gives us a potential root of x = -3/2. However, we must verify that this solution is within the domain of the function, which we determined in Step 1.

Step 4: Verifying the Solution

Recall that the domain restriction for f(x) = -3√(-2x - 3) is x ≤ -3/2. Our potential root, x = -3/2, falls exactly on the boundary of this domain. This means it is a valid solution. To further confirm, we can substitute x = -3/2 back into the original function:

f(-3/2) = -3√(-2(-3/2) - 3) f(-3/2) = -3√(3 - 3) f(-3/2) = -3√0 f(-3/2) = -3 * 0 f(-3/2) = 0

Since f(-3/2) = 0, we have confirmed that x = -3/2 is indeed a root of the function.

Conclusion

Through a systematic approach, we have determined that the function f(x) = -3√(-2x - 3) has one root: x = -3/2. This process involved understanding the function's domain restriction, setting the function to zero, solving for x, and verifying the solution. This comprehensive method is applicable to finding the roots of various types of functions, particularly radical functions. Remember, always consider the domain restrictions when dealing with functions like square roots, as they play a crucial role in determining the validity of solutions.

Graphical Interpretation

The root of a function corresponds to the x-intercept of its graph. In this case, the graph of f(x) = -3√(-2x - 3) intersects the x-axis at x = -3/2. Visualizing the graph can provide a clear understanding of the solution and the function's behavior.

The graphical interpretation of mathematical concepts often provides valuable insights and reinforces understanding. When discussing the roots of a function, the graphical representation helps visualize these roots as the points where the function's graph intersects the x-axis. In other words, the roots are the x-values at which the function's output (y-value) is zero.

For the function f(x) = -3√(-2x - 3), we found that the root is x = -3/2. This means that if we were to plot the graph of this function, it would cross the x-axis at the point (-3/2, 0). The graph's behavior around this point can further illustrate the nature of the root. For instance, we can observe how the function approaches and leaves the x-axis.

To visualize this, consider the general shape of a square root function. The function f(x) = √x starts at the origin (0, 0) and increases gradually as x increases. Our function f(x) = -3√(-2x - 3) is a transformation of this basic square root function. The transformations include:

  1. Reflection about the y-axis: The -2x term inside the square root causes a reflection about the y-axis.
  2. Horizontal compression: The factor of 2 in -2x compresses the graph horizontally.
  3. Horizontal shift: The -3 inside the square root shifts the graph horizontally.
  4. Reflection about the x-axis: The -3 outside the square root reflects the graph about the x-axis.
  5. Vertical stretch: The factor of 3 outside the square root stretches the graph vertically.

These transformations collectively result in a graph that starts at the point (-3/2, 0) and extends to the left. The negative sign outside the square root ensures that the graph lies below the x-axis. This visual representation confirms our algebraic solution and provides a more intuitive understanding of the root.

Furthermore, the graphical perspective highlights the significance of the domain restriction. Since the function is only defined for x ≤ -3/2, the graph exists only to the left of x = -3/2. This reinforces the fact that any solutions outside this interval would be extraneous.

In summary, understanding the graphical interpretation of roots not only validates the algebraic solutions but also enhances our comprehension of the function's overall behavior and characteristics. It’s a powerful tool for both solving and understanding mathematical problems.

Extraneous Solutions

In some cases, squaring both sides of an equation can introduce extraneous solutions. These are solutions that satisfy the transformed equation but not the original equation. It's essential to always check solutions in the original equation to eliminate any extraneous roots. While we verified our solution in this case, this step is crucial in more complex scenarios.

The concept of extraneous solutions is particularly important when dealing with radical equations and rational equations. These are solutions that arise during the solving process but do not satisfy the original equation. They are, in essence, false solutions that need to be identified and discarded to obtain the correct answer.

Extraneous solutions often occur when we perform operations that can alter the solution set of an equation. A common operation that can introduce extraneous solutions is squaring both sides of an equation, as was done in our problem with the square root function. Squaring both sides can turn a false statement into a true statement, thereby creating a solution that didn't exist before.

Consider a simple example to illustrate this point. Suppose we have the equation:

√(x) = -2

This equation has no real solutions because the square root of a number cannot be negative. However, if we square both sides, we get:

(√(x))² = (-2)² x = 4

If we substitute x = 4 back into the original equation, we get:

√(4) = -2 2 = -2

This is a false statement, so x = 4 is an extraneous solution. The correct conclusion is that the original equation has no real solutions.

In the context of our function f(x) = -3√(-2x - 3), we squared both sides of the equation √(-2x - 3) = 0 to eliminate the square root. While we did not encounter an extraneous solution in this case, the process of squaring could potentially introduce one. Therefore, it's crucial to verify the solution by plugging it back into the original equation.

To avoid overlooking extraneous solutions, it's a good practice to:

  1. Solve the equation using standard algebraic techniques.
  2. Substitute each potential solution back into the original equation.
  3. Check if the solution makes the original equation true. If it does not, it's an extraneous solution and should be discarded.

The verification step is a fundamental part of solving radical and rational equations. It ensures that the solutions we obtain are genuine and satisfy the conditions of the original problem. Ignoring this step can lead to incorrect answers and a misunderstanding of the function's behavior.

In summary, the concept of extraneous solutions underscores the importance of careful verification when solving equations, particularly those involving radicals or rational expressions. It’s a crucial step in ensuring the accuracy and validity of the solutions we find.

General Approach

The steps we followed to find the roots of f(x) = -3√(-2x - 3) can be generalized for other functions. This general approach involves:

  1. Determining the domain of the function.
  2. Setting the function equal to zero.
  3. Solving the resulting equation.
  4. Verifying the solutions within the domain.

This systematic method can be applied to various types of functions, making it a valuable tool in mathematical problem-solving. Understanding and applying this approach will enhance your ability to find roots efficiently and accurately.