Problem 34 - Algebra Form and Function

Question

Find the inverse function. $$ q(x)=\frac{4-3 x}{7-9 x} $$

Step-by-Step Solution

Answer

Answer: The inverse function of $$q(x)$$ is $$q^{-1}(x)=\frac{-7x+4}{9x-3}$$.

Step 1: Replace q(x) with 'y'

Replace the function q(x) with the variable 'y' like this: $$ y=\frac{4-3x}{7-9x} $$

Step 2: Swap the roles of 'x' and 'y'

Replace all occurrences of 'y' by 'x' and vice versa: $$ x=\frac{4-3y}{7-9y} $$

Step 3: Solving for 'y'

We need to isolate 'y' on one side of the equation. First, multiply both sides by (7-9y) to get rid of the denominator: $$ x(7-9y)=4-3y $$ Expand the left side of the equation by distributing x across the terms in the parenthesis: $$ 7x-9xy=4-3y $$ Now, move terms involving 'y' to one side and the rest of the terms to the other side: $$ 9xy-3y=-7x+4 $$ Factor out 'y' from the left side of the equation: $$ y(9x-3)=-7x+4 $$ Now, divide by (9x-3) to isolate 'y': $$ y=\frac{-7x+4}{9x-3} $$

Step 4: Rewrite the solution as an inverse function

Finally, rewrite 'y' as the inverse function symbol, q^(-1)(x): $$ q^{-1}(x)=\frac{-7x+4}{9x-3} $$ That's the inverse function: $$ q^{-1}(x)=\frac{-7x+4}{9x-3} $$

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a crucial part of finding an inverse function. It involves rearranging equations and expressions through a series of operations. In the process of finding the inverse of a function like \( q(x)=\frac{4-3x}{7-9x} \), several algebraic steps are taken.
To begin with, the function is set equal to 'y', forming an equation, and the roles of 'x' and 'y' are swapped. This swapping is done because the inverse function effectively reverses the original operations on the input values.
Next, manipulating the equation involves clearing the fraction by multiplying through by the denominator. This elimination of the fraction simplifies the equation, often making it easier to solve for 'y'. For example, multiplying both sides by \( (7-9y) \) in this context clears the fraction:

• \( x(7-9y) = 4-3y \)

Further steps involve expanding terms via distribution, combining like terms, and reorganizing the equation to isolate 'y'. Such steps might include gathering terms with 'y' on one side and factoring out the common variable, as demonstrated in the solution steps:

• Reorganized equation: \( 9xy-3y=-7x+4 \)
• Factoring out 'y': \( y(9x-3)=-7x+4 \)

Rational Functions

Rational functions are functions expressed as the ratio of two polynomials. Here, the original function \( q(x)=\frac{4-3x}{7-9x} \) is a ratio where both the numerator and the denominator are first-degree polynomials in 'x'.
These functions have specific behaviors, such as vertical asymptotes, which occur where the denominator equals zero, making the function undefined. In this function, a vertical asymptote will appear where \( 7-9x=0 \) or \( x=\frac{7}{9} \). Understanding the domain of the function involves knowing these points of discontinuity.
In the context of inverse functions, finding the inverse of a rational function usually involves ensuring that each operation can be reversed correctly, which relies heavily on manipulating both the numerator and denominator separately to revert the function's effect. Recognizing these properties can aid in executing precise algebraic operations as seen in the provided solution.

Function Operations

Function operations refer to the procedures of adding, subtracting, multiplying, dividing, and composing functions, alongside finding their inverses. In dealing with inverses, operations must be reversed in a strictly logical manner.
When deriving the inverse of a function \( q(x) \), the goal is to identify a function \( q^{-1}(x) \) such that \( q(q^{-1}(x))=x \). The process begins by executing reverse operations on the original function by substituting 'y' and swapping 'x' with 'y'.
The algebraic steps involved in isolating 'y' like multiplying through by a denominator, distributing terms, and factoring are typical operations used to simplify and manipulate the function. This paves the way to obtaining the inverse expression, \( q^{-1}(x)=\frac{-7x+4}{9x-3} \), confirming that the intended operations are reversed accurately.Employing these function operations ensures that the inverse exactly undoes what the original function does, demonstrating a key property of inverse functions: their ability to revert outputs back into inputs.