Question

How would you begin to rewrite the function \(g(x)=\frac{4 x+1}{x+2}\) to obtain the form \(g(x)=\frac{a}{x-h}+k ?\) (A) \(g(x)=\frac{4(x+2)-7}{x+2}\) (B) \(g(x)=\frac{4(x+2)+1}{x+2}\) (C) \(g(x)=\frac{(x+2)+(3 x-1)}{x+2}\) (D) \(g(x)=\frac{4 x+2-1}{x+2}\)

Short answer

None of the given options correspond to the form \(g(x)=\frac{a}{x-h}+k\).

Step-by-step solution

Consider Option A

By expanding the numerator of option (A) we find: \[4(x+2)-7 = 4x +8 - 7 = 4x + 1 \] which is the same as the original function. Hence, option (A) merely maintains the same function but expressed in a different way. This doesn't represent the form we are attempting to obtain.

Consider Option B

By expanding the numerator of option (B) we find: \[4(x+2)+1 = 4x +8 + 1 = 4x + 9 \] which incorporates a constant different from the original function. This suggests that this isn't the correct option.

Consider Option C

By expanding the numerator of option (C) we find: \[(x+2)+(3x-1) = 4x + 1 \] which represents the original function, but just as in option (A), doesn't represent the form we seek.

Consider Option D

By considering the numeration of option (D) we find: \[4x + 2 - 1 = 4x + 1 \] which again is the same as our original function. It doesn't satisfy the form we are after.

Key concepts

Transforming Functions

Transforming functions involves changing the form of a function while keeping its overall behavior intact. This may be done to make calculations easier, to reveal new insights, or to match a desired form, as in this exercise.

For rational functions, these transformations often involve manipulating the numerator or denominator. This might mean expanding, factoring, or completing the square. The goal is to arrive at a form that highlights certain features of the function, like asymptotes, intercepts, or shifts.

With the function \(g(x) = \frac{4x+1}{x+2}\), the task is to rewrite it in the form \(g(x) = \frac{a}{x-h} + k\). This common rational function form reveals horizontal and vertical shifts, helping us understand its graph more clearly.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions using the rules of algebra. This includes distributing factors, combining like terms, and canceling terms when possible.

In the given exercise, algebraic manipulation is used to explore different ways to write the function \(g(x) = \frac{4x+1}{x+2}\). Each option involves altering the numerator to potentially match a target form.

For example, if we take option (A), \(4(x+2) - 7\) becomes \(4x + 1\) after expanding:

• The distributed term, \(4(x+2)\), expands to \(4x + 8\)
• Subtracting 7 from 8 results in \(1\), matching the original numerator

By applying similar manipulations to each option, we assess which accurately transforms the original function into the desired form.

Function Rewriting

Function rewriting is a specific application of algebraic manipulation where we change the form of a function to suit a given purpose. This might involve rewriting a function to make graphing easier or to compare it against a standard or ideal form.

In this exercise, the goal is to rewrite \(g(x) = \frac{4x+1}{x+2}\) into \(\frac{a}{x-h} + k\). This requires identifying parts of the function that resemble components of the new form, such as isolating terms to represent horizontal and vertical translations.

Options for rewriting the function suggest different ways of splitting the numerator and are evaluated for accuracy in reaching the desired format. The exploration includes expanding expressions: for instance, option (B) uses \(4(x+2) + 1 = 4x + 9\), which is incorrect since it doesn’t match our translation goals.

Algebra II

Algebra II is a foundational high school math course that covers advanced algebra concepts, including rational functions, transformations, and complex expressions.

In Algebra II, students learn how to manipulate functions and work with various forms of algebraic expressions to understand their properties and behavior. This exercise taps into such skills by requiring the rewriting of functions and testing logical reasoning through algebraic adjustments.

Mastery of topics like

• expansions,
• factoring,
• equations of the form \(\frac{a}{x-h} + k\),

ensures students can transition from the classroom to real-world applications. The step-by-step solution provided reflects the critical thinking and procedural skills central to Algebra II, showing how transformation and rewriting roles play in functional analysis.