Problem 30 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

How would you begin to rewrite the function \(g(x)=\frac{x}{x-5}\) to obtain the form \(g(x)=\frac{a}{x-h}+k ?\) (A) \(g(x)=\frac{x(x+5)(x-5)}{x-5}\) (B) \(g(x)=\frac{x-5+5}{x-5}\) (C) \(g(x)=\frac{x}{x-5+5}\) (D) \(g(x)=\frac{x}{x}-\frac{x}{5}\)

Step-by-Step Solution

Answer

The answer is (B) \(g(x)=\frac{x-5+5}{x-5}\)

Step 1: Analyze the Candidates

First, evaluate each of the candidate equations to determine if they equate to \(g(x)=\frac{x}{x-5}\). Let's start with the first candidate, \(g(x)=\frac{x(x+5)(x-5)}{x-5}\) . Simplifying this equation will reveal that it does not compute to the original function. Continue this process with each of the remaining candidates.

Step 2: Simplify the Function Candidates

Simplifying the second candidate, \(g(x)=\frac{x-5+5}{x-5}\), gives \(g(x)=\frac{x}{x-5}\), which is identical to the original function. Hence this is the correct choice. No need to check the remaining options as we have found the correct answer.

Step 3: Verify the Correct Candidate

Although we've found the answer after step 2, it's always a good practice to confirm. Checking the remaining two candidates also results in different functions from the original one, confirming that the answer obtained in step 2 is correct.

Key Concepts

Rational Function Simplification

When tackling the simplification of rational functions, it's essential to look at the function's numerator and denominator. A rational function is one that can be expressed as the ratio of two polynomials. In the case of the exercise where we have the function \(g(x)=\frac{x}{x-5}\), our first impulse might be to perform operations that will help us find the equivalent function in the form of \(g(x)=\frac{a}{x-h}+k\).

The process involves breaking down the function into simpler parts, potentially by adding or subtracting terms in the numerator and then simplifying. For example, in option (B) \(g(x)=\frac{x-5+5}{x-5}\), adding and then subtracting 5 seems counterintuitive, but it's actually a strategic move to transform the function into a desirable format after simplification. This strategic adding and subtracting within a rational function can often lead to a clearer understanding of its characteristics, such as vertical asymptotes or x-intercepts.

Simplifying a rational function correctly requires attention to the relationship between the numerator and the denominator, aiming to maintain the function's integrity. This is where knowledge in factoring and canceling like terms is invaluable, ensuring that the simplified function is equivalent to the original.

Mathematical Function Transformation

Mathematical function transformation is about altering the basic graph of a function to achieve a new graph that reflects the changes in the equation. It includes shifting, stretching, compressing, and reflecting the graph. To rewrite the rational function \(g(x)=\frac{x}{x-5}\) into the desired form of \(g(x)=\frac{a}{x-h}+k\), we are considering transforming the given function.

The actual transformation involves a horizontal shift, represented by \(h\), and a vertical shift, depicted by \(k\). These two parameters are crucial as they dictate how the graph of the function will be translated along the x-axis and y-axis, respectively. To achieve the transformation, we can add and then subtract the same value in the numerator to form a term that would give us the ‘h’ value when set to zero, and the constant outside to derive the ‘k’ value. This is not exactly apparent in the exercise options provided, which subtly indicates the importance of practicing various transformation techniques. Through transformation, we gain a more profound understanding of the behavior and the graphical representation of functions.

Algebraic Expressions

Algebraic expressions are the cornerstone of understanding and manipulating functions in algebra. These expressions are made up of variables, numbers, and operators (like +, -, *, /). The ability to move fluidly between different forms of algebraic expressions is what enables one to solve a wide variety of mathematical problems.

In the context of our exercise, the expression \(g(x)=\frac{x}{x-5}\) is being analyzed to determine a correct transformation that leads to a predetermined form. The manipulation of algebraic expressions here involves adding zero in a strategic form, such as \(5-5\), or factoring and expanding polynomials without altering the value of the expression. A deeper understanding of algebraic expressions allows students to engage with the material in a more versatile way, and utilize properties of algebraic operations to simplify or modify expressions to better suit the form that is needed. This kind of manipulation is vital for function transformation and simplifying expressions in not only textbook exercises, but in applying math to real world problems.