Question

If \(f(x)=\frac{2 x-7}{5 x+1},\) find \(f^{-1}(x)\).

Short answer

f^{-1}(x) = \frac{-x - 7}{5x - 2}

Step-by-step solution

Set up the equation for inversion

Start by setting the function equal to a new variable. Let \[ y = \frac{2x - 7}{5x + 1} \]

Swap variables

Switch the roles of x and y in the equation to begin finding the inverse. This gives \[ x = \frac{2y - 7}{5y + 1} \]

Clear the fraction

Multiply both sides of the equation by the denominator to clear the fraction. \[ x(5y + 1) = 2y - 7 \]

Distribute and rearrange

Distribute x on the left side and rearrange the equation to isolate all terms involving y on one side. \[ 5xy + x = 2y - 7 \] Subtract 2y from both sides and subtract x from both sides: \[ 5xy - 2y = -x - 7 \]

Factor out y

Factor out y from the left side of the equation. \[ y(5x - 2) = -x - 7 \]

Solve for y

Divide both sides of the equation by (5x - 2) to solve for y. \[ y = \frac{-x - 7}{5x - 2} \]

Verify and write the inverse function

The inverse function, denoted as \(f^{-1}(x)\), is thus \[ f^{-1}(x) = \frac{-x - 7}{5x - 2} \]

Key concepts

inverse function

An inverse function essentially reverses the effect of the original function. If you have a function, denoted as \( f(x) \), its inverse, denoted as \( f^{-1}(x) \), will give you back the original input when applied to the output of the original function.
For example, if \( y = f(x) \) then \( x = f^{-1}(y) \). To find the inverse, we sometimes need to follow several steps outlined clearly.
Let's explore what it means to find an inverse function, using the given exercise as an example.

The original function is \( f(x) = \frac{2x - 7}{5x + 1} \). To find the inverse:

• First, we set the function equal to \( y \) so that we can work with the equation: \( y = \frac{2x - 7}{5x + 1} \).


• Next, we swap the roles of \( x \) and \( y \) because the inverse function does the opposite of the original function: \( x = \frac{2y - 7}{5y + 1} \).
• Then, we clear the fraction by multiplying both sides by \( 5y + 1 \).
• From there, we isolate \( y \) by rearranging the equation and solving for all terms involving \( y \).


• Once isolated, we factor out \( y \) and solve for it to find the inverse function: \( f^{-1}(x) = \frac{-x - 7}{5x - 2} \).

algebra

Solving for inverse functions requires a solid understanding of algebra. Algebra involves manipulating mathematical symbols to solve for unknown variables.
In this exercise, multiple algebraic techniques were used:

• Firstly, variables were swapped to represent the role change from function to inverse function.


• To clear the fraction, multiplication was employed, which is a fundamental algebraic operation.
• Rearrangement of terms required adding and subtracting like terms on both sides of the equation.
• Factoring, another key algebraic technique, was used to isolate the variable \( y \).
• Finally, dividing by the coefficient of the isolated term helped to solve for the variable \( y \).

Each of these steps is part of a broader algebraic toolkit that assists in navigating the process of finding inverse functions. Practicing these techniques will make solving similar problems more intuitive over time.

rational functions

Rational functions are quotients of polynomials. They can often be complicated to work with because they involve a variable in the denominator. In this problem, the function \( f(x) = \frac{2x - 7}{5x + 1} \) is a rational function.

Characteristics of rational functions include:

• A numerator that can be a polynomial (in this case, \( 2x - 7 \)).
• A denominator that is also a polynomial (in this case, \( 5x + 1 \)).

When finding the inverse of a rational function:

• The variable swap may result in a complex equation initially.
• Clearing the fraction by multiplying both sides by the denominator is often a necessary step.
• You need to handle the components of both polynomials carefully to isolate the variable representing the inverse function.

Understanding how to manipulate these polynomials correctly is crucial for solving the problem and finding the correct inverse.

solving equations

Solving equations is a fundamental skill in mathematics that extends into finding inverse functions. In the given exercise, several steps involved solving an equation to isolate and determine the inverse function.
Here are the key solving techniques used in this exercise:

• Starting with an initial equation setup \( y = \frac{2x - 7}{5x + 1} \), you must switch variables to represent the inverse scenario \( x = \frac{2y - 7}{5y + 1} \).


• Next, to eliminate the fraction, multiply both sides by the denominator \( 5y + 1 \) which gives \( x(5y + 1) = 2y - 7 \).
• Distribute and then rearrange the terms to bring all \( y \)-related terms on one side, resulting in a combined equation.


• Factor out the common variable \( y \) to simplify the equation further.
• Finally, solve for \( y \) by dividing both sides by the remaining coefficient to find \( y = \frac{-x - 7}{5x - 2} \).

Each of these steps is critical in transitioning from the original function to its inverse and showcases the importance of methodical problem-solving techniques in algebra.