Question

For each rational function given, determine the inverse function, \(f^{-1}(x)\) a) \(f(x)=\frac{x-3}{x+1}\) b) \(f(x)=\frac{2 x}{x-5}+4\)

Short answer

a) \text{\(f^{-1}(x) = \frac{-x - 3}{x - 1}\)} b) \text{\(f^{-1}(x) = \frac{5x - 20}{x - 6}\)}

Step-by-step solution

Interchange variables

To find the inverse of a function, interchange the variables. For both parts (a) and (b), replace every instance of 'y' with 'x' and every instance of 'x' with 'y'.

Express 'y' in terms of 'x' for part (a)

For the function in part (a): 1. Start with the equation: \text{\(y = \frac{x - 3}{x + 1}\)}2. Interchange 'x' and 'y': \text{\(x = \frac{y - 3}{y + 1}\)}

Solve for 'y' in part (a)

Solve for 'y': \text{\(x(y + 1) = y - 3\)}\text{\(xy + x = y - 3\)}\text{\(xy - y = - x - 3\)}\text{\(y(x - 1) = - x - 3\)}\text{\(y = \frac{-x - 3}{x - 1}\)} Thus, the inverse function for part (a) is: \text{\(f^{-1}(x) = \frac{-x - 3}{x - 1}\)}

Express 'y' in terms of 'x' for part (b)

For the function in part (b): 1. Start with the equation: \text{\(y = \frac{2x}{x - 5} + 4\)}2. Interchange 'x' and 'y': \text{\(x = \frac{2y}{y - 5} + 4\)}

Solve for 'y' in part (b)

Solve for 'y': First, subtract 4 from both sides: \text{\(x - 4 = \frac{2y}{y - 5}\)} Then, multiply both sides by \text{\(y - 5\)}: \text{\((x - 4)(y - 5) = 2y\)} Expand and arrange terms: \text{\(xy - 5x - 4y + 20 = 2y\)} Combine 'y' terms: \text{\(xy - 4y - 2y = 5x - 20\)} \text{\(y(x - 6) = 5x - 20\)} Solve for 'y': \text{\(y = \frac{5x - 20}{x - 6}\)} Thus, the inverse function for part (b) is: \text{\(f^{-1}(x) = \frac{5x - 20}{x - 6}\)}

Key concepts

Rational Functions

A rational function is a type of function represented by a ratio of two polynomials. An example is \(\frac{x-3}{x+1}\). In these functions, the numerator and the denominator are both polynomials. Key characteristics include:

• They can have vertical asymptotes where the denominator is zero.
• They can exhibit horizontal asymptotes depending on the degrees of the numerator and denominator.
• They might include holes (points where both the numerator and denominator are zero).

Understanding these properties is essential when manipulating and finding inverse functions of rational forms.

Inverse Operations

Inverse operations are used to reverse the effects of another operation. For instance, addition is the inverse of subtraction. The primary inverse operations involved in finding inverse functions include:

• Addition and subtraction
• Multiplication and division

Finding the inverse of a function means solving for the original input variable (y) in terms of the output variable (x). For example, with \(\frac{x-3}{x+1}\), we swapped x and y to \(x = \frac{y-3}{y+1}\), and then solved for y to find \(\frac{-x-3}{x-1}\).

Function Manipulation

Function manipulation involves algebraic techniques to rearrange and simplify equations. When finding an inverse:

• First, interchange x and y.
• Next, solve for the new variable y.
• Simplify step-by-step by isolating y.
• Perform operations like multiplication, division, addition, and subtraction to both sides equally to isolate the variable.

As seen in the example \(y = \frac{2x}{x-5} + 4\), we substituted x for y and methodically isolated y, reaching \(\frac{5x-20}{x-6}\) as the inverse.