Question

Match each equation to the single function that can be used to solve it graphically. a) \(\frac{x}{x-2}+6=x\) b) \(6-x=\frac{x}{x-2}+2\) c) \(6-\frac{x}{x-2}=x-2\) d) \(x+6=\frac{x}{x-2}\) \(\mathbf{A} \quad y=\frac{x}{x-2}+x-8\) \(\mathbf{B} \quad y=\frac{x}{x-2}-x+6\) \(\mathbf{c} \quad y=\frac{x}{x-2}-x-6\) \(\mathbf{D} \quad y=\frac{x}{x-2}+x-4\)

Short answer

a) B, b) D, c) A, d) C

Step-by-step solution

- Simplify Each Given Equation

Each of the four given equations needs to be simplified. Simplifying will help identify the correct function that represents each equation.

- Simplify Equation (a)

For equation \(\frac{x}{x-2}+6=x\), subtract x from both sides: \(\frac{x}{x-2} + 6 - x = 0\). This simplifies to the function \(\frac{x}{x-2} - x + 6\), which matches function B.

- Simplify Equation (b)

For equation \(6-x = \frac{x}{x-2}+2\), rearrange to get: \(\frac{x}{x-2} + 2 + x - 6 = 0\). This simplifies to the function \(\frac{x}{x-2} + x - 4\), which matches function D.

- Simplify Equation (c)

For equation \(6-\frac{x}{x-2} = x-2\), rearrange to get: \(\frac{x}{x-2} + x - 8 = 0\). This simplifies to the function \(\frac{x}{x-2} + x - 8\), which matches function A.

- Simplify Equation (d)

For equation \(x+6 = \frac{x}{x-2}\), subtract \( \frac{x}{x-2} \) from both sides to get \( x + 6 - \frac{x}{x-2} = 0 \). This simplifies to the function \( \frac{x}{x-2} - x - 6 \), which matches function C.

Key concepts

Function Simplification

Simplifying functions means breaking down complex equations into simpler forms. This makes them easier to work with. Let's take a look at an example:
Given the equation \(\frac{x}{x-2} + 6 = x\), we want to make it simpler.
To do this, subtract x from both sides: \( \frac{x}{x-2} + 6 - x = 0 \).
This rearranges the terms and simplifies the equation. Now it looks like \( \frac{x}{x-2} - x + 6 \).
Each term is easier to understand and manage. This is the process of function simplification.

Here are the steps in brief:

• Identify the complex equation.
• Perform operations to isolate variables.
• Combine and simplify terms.

Function simplification is a powerful tool for solving mathematical problems.

Solving Equations Graphically

Sometimes, equations are easier to solve when you use graphs. Graphical methods provide visual insights.
By plotting a function, you can see intercepts, slopes, and lines of symmetry. Consider two functions

• y = f(x)
• y = g(x)

To find the solution, plot these functions on a graph. The points where the lines intersect are the solutions.
These points represent the values of x that satisfy both equations.

In our exercise, we use graphical methods to match equations with their respective functions. For example, \( y = \frac{x}{x-2} - x + 6 \) represents our simplified function. Plotting it helps us visually confirm intersections and solutions.

Graphical solutions offer a clear understanding of complex equations. They allow you to see real-time changes and outcomes.

Rational Functions

Rational functions are equations that involve fractions. They typically look like this: \( \frac{P(x)}{Q(x)} \), where P(x) and Q(x) are polynomials.
A common feature of rational functions is their domain. The domain is all possible input values (x-values). Often, Q(x) restricts these values because the function becomes undefined where Q(x) is zero.

For example, \( \frac{x}{x-2} \) is undefined at x = 2. This is because dividing by zero is not possible.

When working with rational functions:

• Find the domain by setting Q(x) ≠ 0.
• Simplify the function if possible.
• Consider vertical asymptotes and horizontal limits.

Rational functions often model real-world situations. Mastering them helps you tackle a variety of problems effectively.

Equation Rearrangement

Rearranging equations is a crucial skill. It means changing the equation's format without changing its value.

Let's take equation (b) from our exercise: \( 6 - x = \frac{x}{x-2} + 2 \). To simplify this, we need to rearrange it:

Add x to both sides to isolate the rational function: \( 6 - x + x = \frac{x}{x-2} + 2 + x \)
Simplify it to: \( \frac{x}{x-2} + 2 + x - 6 = 0 \)

Rearrangement uses basic algebra rules:

• Addition and Subtraction
• Multiplication and Division
• Combining like terms

With practice, equation rearrangement becomes easier. It transforms complex problems into more manageable forms, paving the way for further simplification and solution.