Inverse Of M(x) = X^2 - 17x Explained With Domain Restriction

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The concept of inverse functions is fundamental in mathematics, particularly in algebra and calculus. An inverse function essentially "undoes" the original function. In simpler terms, if a function f takes an input x and produces an output y, the inverse function f⁻¹ takes y as input and returns x. This article delves into the intricacies of finding the inverse of the quadratic function m(x) = x² - 17x, with a particular focus on the domain restriction x ≥ 17/2. We will explore the step-by-step process of finding the inverse, discuss the importance of domain restrictions, and provide a comprehensive explanation of the resulting inverse function.

Finding the Inverse of m(x) = x² - 17x

To find the inverse of a function, we typically follow a series of steps. Let's apply these steps to the function m(x) = x² - 17x.

Step 1: Replace m(x) with y

The first step is to replace the function notation m(x) with y. This makes the equation easier to manipulate algebraically. So, we rewrite the function as:

y = x² - 17x

Step 2: Swap x and y

The next step is crucial in finding the inverse. We swap the variables x and y. This reflects the fundamental idea of an inverse function – reversing the roles of input and output. After swapping, we get:

x = y² - 17y

Step 3: Solve for y

Now, the challenge is to isolate y on one side of the equation. This often involves algebraic manipulation. In this case, we have a quadratic equation in terms of y. To solve it, we can use the method of completing the square.

Completing the Square

Completing the square is a technique used to rewrite a quadratic expression in a form that includes a perfect square trinomial. This makes it easier to solve for the variable. Here's how we apply it to our equation:

  1. Rewrite the equation:

    x = y² - 17y

  2. Add and subtract (17/2)²:

    To complete the square, we need to add and subtract the square of half the coefficient of the y term. The coefficient of the y term is -17, so half of it is -17/2, and its square is (17/2)² = 289/4. Adding and subtracting this value doesn't change the equation:

    x + 289/4 - 289/4 = y² - 17y

  3. Rearrange to form a perfect square trinomial:

    We rearrange the terms to group the perfect square trinomial:

    x + 289/4 = y² - 17y + 289/4

  4. Factor the perfect square trinomial:

    The right side of the equation is now a perfect square trinomial, which can be factored as:

    x + 289/4 = (y - 17/2)²

Step 4: Isolate y

Now we can proceed to isolate y:

  1. Take the square root of both sides:

    Taking the square root of both sides gives us:

    ±√(x + 289/4) = y - 17/2

  2. Solve for y:

    Add 17/2 to both sides:

    y = 17/2 ± √(x + 289/4)

Step 5: Write the Inverse Function

We now have two possible solutions for y, which represent the two possible branches of the inverse function:

m⁻¹(x) = 17/2 + √(x + 289/4)

m⁻¹(x) = 17/2 - √(x + 289/4)

The Significance of the Domain Restriction x ≥ 17/2

The original function m(x) = x² - 17x is a parabola. Without any restrictions, a parabola is not a one-to-one function, meaning it does not pass the horizontal line test. A function must be one-to-one to have a unique inverse. To ensure that m(x) has an inverse, we need to restrict its domain.

The domain restriction x ≥ 17/2 effectively considers only the right half of the parabola. This restriction makes the function one-to-one, allowing us to define a unique inverse.

Impact on the Inverse Function

The domain restriction x ≥ 17/2 for the original function has implications for the range of the inverse function. When x ≥ 17/2 for m(x), the range is m(x) ≥ -289/4. This range becomes the domain of the inverse function, m⁻¹(x).

Given the domain restriction x ≥ 17/2, the correct inverse function is:

m⁻¹(x) = 17/2 + √(x + 289/4)

This is because the domain restriction ensures that we are considering the portion of the parabola where x is greater than or equal to 17/2, which corresponds to the right half of the parabola. Consequently, the inverse function must yield values greater than or equal to 17/2.

Analyzing the Inverse Function m⁻¹(x) = 17/2 - √(x + 289/4)

The statement in the prompt suggests that the inverse function with the domain restriction x ≥ 17/2 is:

m⁻¹(x) = 17/2 - √(x + 289/4)

Let's analyze why this is not the correct inverse function for the given domain restriction.

Evaluating the Inverse Function

Consider the original function m(x) = x² - 17x with the domain restriction x ≥ 17/2. The vertex of the parabola is at x = 17/2, and the function opens upwards. Therefore, for x ≥ 17/2, the function values are greater than or equal to the value at the vertex.

The x-coordinate of the vertex is x = 17/2. Plugging this into m(x), we get:

m(17/2) = (17/2)² - 17(17/2) = 289/4 - 289/2 = -289/4

So, the range of m(x) for x ≥ 17/2 is m(x) ≥ -289/4. This means the domain of the inverse function m⁻¹(x) is x ≥ -289/4.

Now, let's consider the proposed inverse function:

m⁻¹(x) = 17/2 - √(x + 289/4)

For this function to be the inverse, it must satisfy the condition that when x ≥ -289/4, the values of m⁻¹(x) must be greater than or equal to 17/2 (due to the domain restriction on the original function).

However, the negative sign in front of the square root term indicates that the values of m⁻¹(x) will be less than or equal to 17/2. This contradicts the domain restriction x ≥ 17/2 for the original function.

Example

Let's take an example to illustrate this. Consider x = -289/4:

m⁻¹(-289/4) = 17/2 - √(-289/4 + 289/4) = 17/2 - √0 = 17/2

Now, consider a value slightly greater than -289/4, say x = -288/4 = -72:

m⁻¹(-72) = 17/2 - √(-72 + 289/4) = 17/2 - √(1/4) = 17/2 - 1/2 = 16/2 = 8

Here, m⁻¹(-72) = 8, which is less than 17/2. This demonstrates that the proposed inverse function m⁻¹(x) = 17/2 - √(x + 289/4) does not satisfy the condition that its values must be greater than or equal to 17/2 for the given domain restriction.

The Correct Inverse Function

The correct inverse function, given the domain restriction x ≥ 17/2, is:

m⁻¹(x) = 17/2 + √(x + 289/4)

This function ensures that the values of m⁻¹(x) are always greater than or equal to 17/2, satisfying the domain restriction of the original function.

Conclusion

In summary, finding the inverse of a function involves several steps, including swapping variables and solving for the new dependent variable. For the function m(x) = x² - 17x, completing the square is a crucial technique to isolate y. Domain restrictions play a vital role in ensuring that a function has a unique inverse. In the case of m(x), the restriction x ≥ 17/2 makes the function one-to-one and allows us to define its inverse uniquely. The correct inverse function for the given domain restriction is m⁻¹(x) = 17/2 + √(x + 289/4), as it ensures that the inverse function's values are consistent with the original function's domain restriction. Understanding these concepts is essential for mastering inverse functions and their applications in mathematics.