The range of a rational function is the set of all outputs (y-values) that it produces. To find the range of a rational function y= f(x):
- If we have f(x) in the equation, replace it with y.
- Solve the equation for x.
- Set the denominator of the resultant equation ≠ 0 and solve it for y.
- Set of all real numbers other than the values of y mentioned in the last step is the range.
Example: Find the range of f(x) = (2x + 1) / (3x – 2).
Solution:
Let us replace f(x) with y. Then y = (2x + 1) / (3x – 2). Now, we will solve this for x.
(3x – 2) y = (2x + 1)
3xy – 2y = 2x + 1
3xy – 2x = 2y + 1
x (3y – 2) = (2y + 1)
x = (2y + 1) / (3y – 2)
Now (3y – 2) ≠ 0
y ≠ 2/3
So the range = {y ∈ R | y ≠ 2/3}