- It is useful to know the definition of rational expression because it helps you recognize and manipulate these types of expressions in Algebra like factoring, simplifying, solving equations, or graphing. It also helps you avoid dividing by zero and find restrictions and predict the behavior of functions.
Example: f(x) = (x² - 4)/(x - 2)
Knowing this is rational tells us we need to check for restrictions and that we might be able to simplify it by factoring: f(x) = (x + 2)(x - 2)/(x - 2) = x + 2 (with restriction x ≠ 2).
- We state restrictions because division by zero is undefined in math.
We state restrictions immediately when we come across a rational expression before doing any operations. To find it, set the denominator equal to zero and solve for the variable. Restrictions appear as vertical asymptotes or removable discontinuities where the function is undefined.
Example: For f(x) = 3/(x - 5), set x - 5 = 0, so x = 5 is restricted. This creates a vertical asymptote at x = 5.
- Rational functions have unique characteristics:
- Asymptotes: Vertical lines (where denominator = 0) and horizontal lines
- Discontinuities: Breaks in the graph where the function is undefined
- End behavior: Approaches asymptotes rather than continuing infinitely in one direction
Example: f(x) = 1/x has a vertical asymptote at x = 0 and horizontal asymptote at y = 0.
Contrast with other functions:
- Linear functions--> Continuous straight lines with no breaks, Quadratic functions-->Smooth parabolas with no asymptotes, Exponential functions -->Continuous curves that approach but never touch horizontal asymptotes
- Simplifying first is important because working with simpler expressions reduces errors, helps identify which restrictions are "real" vs. just from factored forms, and shows holes vs. asymptotes correctly
- The key difference is that holes represent missing points whereas asymptotes represent boundaries the function approaches but never reaches.
