A rational function is a ratio of two polynomials \(\frac{P(x)}{Q(x)}\). Understanding its domain and asymptotes is crucial for graphing and analysis.
The domain includes all real numbers except where \(Q(x) = 0\)
To find excluded values, factor \(Q(x)\) and solve \(Q(x) = 0\)
Each x-value that makes \(Q(x) = 0\) creates a vertical asymptote, unless it's also a root of \(P(x)\)
For \(f(x) = \frac{x^2 - 4}{x - 2}\):
Factor numerator: \(\frac{(x+2)(x-2)}{x-2}\)
Cancel common factor: \(x + 2\)
Domain: \(x \neq 2\)
Note: This creates a hole at \(x = 2\)
Horizontal asymptotes describe the end behavior of rational functions as \(x\) approaches \(\pm\infty\).
If degree of \(P(x)\) < degree of \(Q(x)\):
Horizontal asymptote at \(y = 0\)
Example: \(f(x) = \frac{2x}{x^2 + 1}\) has HA: \(y = 0\)
If degree of \(P(x)\) = degree of \(Q(x)\):
HA at \(y = \frac{\text{leading coefficient of }P(x)}{\text{leading coefficient of }Q(x)}\)
Example: \(f(x) = \frac{3x^2 + 1}{x^2 - 2}\) has HA: \(y = 3\)
If degree of \(P(x)\) > degree of \(Q(x)\):
No horizontal asymptote
Function grows without bound
To graph rational functions, follow these steps:
Find domain (excluded values)
Find vertical asymptotes
Find horizontal/oblique asymptotes
Find x and y-intercepts
Find any holes
Plot additional points and sketch the curve
Key features:
Domain: \(x \neq 1\)
Vertical asymptote: \(x = 1\)
Horizontal asymptote: \(y = x\)
Hole at \((1, 2)\)
Piecewise rational functions combine different rational expressions over different intervals.
Key concepts:
Finding domain for each piece
Identifying discontinuities at transition points
Analyzing end behavior for each piece
Analyzing behavior near transitions
Key features:
Discontinuity at \(x = 1\)
Vertical asymptotes at \(x = 1\) and \(x = 2\)
Different end behaviors for each piece
Domain: \((-\infty,1) \cup (1,2) \cup (2,\infty)\)
Test your knowledge of all rational function concepts.