Solving Rational Functions

Rational functions are fractions where the numerator and denominator are both polynomials. To solve equations involving rational functions, we often need to find values that make the function equal to zero or undefined. The key steps include finding a common denominator, simplifying the equation, and solving for the variable.

Worked Example

Problem: Solve the equation $\frac{x+2}{x-3} = 1$.

Step 1: Set the equation to zero by subtracting 1 from both sides: $\frac{x+2}{x-3} - 1 = 0$

Step 2: Find a common denominator (which is $x-3$): $\frac{x+2 - (x-3)}{x-3} = 0$

Step 3: Simplify the numerator: $\frac{x+2 - x + 3}{x-3} = 0$ $\frac{5}{x-3} = 0$

Step 4: To find when a fraction equals zero, set the numerator equal to zero: $5 = 0$

This has no solution. Therefore, there are no values of $x$ that satisfy the original equation.

Step 5: Check the denominator to ensure it is not zero:

eq 0 \\Rightarrow x eq 3$$ Thus, the solution set is empty, and $x = 3$ is a restriction. ## Key Questions - What does it mean for a rational function to be undefined? - How can you find the vertical asymptotes of a rational function? - What steps would you take to solve the equation $$\frac{2x}{x+1} = 3$$? - Explain how to determine the domain of a rational function. - Why is it important to check for extraneous solutions after solving? ## Keywords - Rational function - Denominator - Common denominator - Asymptote - Extraneous solution