Solving for Unknowns in Combined Functions: A Deep Dive

When dealing with combined mathematical functions, it's crucial to understand how to manipulate and solve for unknowns effectively. This article will explore a detailed solution for a specific problem, including the application of the quadratic equation, geometric transformations, and the behavior of combined functions.

Understanding the Problem

The problem at hand involves solving for unknowns in a combined function. We start by examining the individual components and then combine them to form a more complex equation. Let's break down each step.

Part A: Simplifying and Solving the Equation

solving √x12√x-12 x2 using the Pythagorean Theorem

The given equation is √(x12)√(x-12) x2. To solve this, we first recognize the relationship between the square roots and the unknowns.

Simplifying the Expression:

√x12 √x - 12 x2

Recognizing the Pythagorean Theorem:

a2 b2 c2

Substituting the Values:

x12 x - 12 x2

Combining Like Terms:

x2 - 2x - 2 0

Now, we solve this quadratic equation using the quadratic formula, {[-b ± √(b2 - 4ac)] / 2a} where a 1, b -2, and c -2.

x {[-(-2) ± √((-2)2 - 4(1)(-2)))] / 2(1)}

Calculating the Roots:

x {2 ± √(4 8)}/2

x 2 ± √12 / 2

x 2 ± 2√3 / 2

x 1 ± √3

Discarding the Negative Root:

Since we are looking for a positive solution, we discard x 1 - √3.

Thus, x 1 √3.

Part B: A Similar Equation

The second part of the problem follows a similar pattern.

Start with the Equation:

x - 2√x√(y) - 1 2xy

Simplify the Equation:

2x√y 2xy

x√y xy

x xy - yx

x yx - yx

Analysis of Combined Functions

Formulating Combined Functions

Combining functions involves understanding how geometric transformations and algebraic manipulations interact. Consider the two functions:

Function f: A simple straight line with gradient a and intercept b.

Function g: A hyperbola offset from the y-axis by -d/c, passing through 0, d.

Behavior of the Combined Functions

To analyze the combined function, we sum the two functions. The combined function looks like a skewed hyperbola. We can find its intercept and solve for its roots and turning points.

Intercept and Roots:

The intercept of the combined function is given by b1/d.

To find the roots, we solve:

ax(b)(1/cx - d) 0

This equation simplifies to:

acx^2 - bcx - ad 0

For this quadratic equation to have real roots, the discriminant bc - ad^2 ≥ 4ac.

Derivative and Turning Points:

The derivative of the combined function is:

a - c / (c x d^2)

Setting the derivative to zero, we find the turning points:

x ±√(1 / (ac)) - d/c

Sketching the Graphs

To sketch the combined function, we need specific values for a, b, c, and d. Given ac is negative, we find no turning points but two roots, and the intercept is 3.

A detailed graph would show the combined function approaching the straight line for large values of x and the hyperbola for values near the asymptote.

Conclusion

Solving for unknowns in combined functions requires a thorough understanding of algebraic manipulations, the use of the quadratic equation, and geometric transformations. By breaking down the problem into manageable steps, we can effectively arrive at the solution and understand the behavior of the combined functions.

Now that you have an in-depth understanding, you can apply these techniques to similar problems involving combined functions.