Rational and Logarithmic Functions Reference

Rational Functions and Asymptotes

A rational function is a fraction of polynomials. The denominator controls restrictions and vertical behavior, while the degrees of the numerator and denominator guide end behavior.

Solved Examples

Jump directly to a worked example by problem type.

Fact Table

Question Where to Look Result
What values are not in the domain? Original denominator. Set the denominator not equal to zero.
Where are the vertical asymptotes? Denominator after canceling common factors. Uncanceled denominator zeros give vertical asymptotes.
Where are the holes? Factors that cancel. The canceled x-value is a hole, not an asymptote.
What is the horizontal asymptote? Degrees of numerator and denominator. Compare degrees before long division.
Where are the x-intercepts? Simplified numerator. Zeros of the numerator, unless the factor canceled.
Where is the y-intercept? Evaluate $f(0)$. Use only if $0$ is in the domain.

Content Formulas

Rational Function
$$f(x)=\frac{p(x)}{q(x)},\quad q(x)\ne0$$
Domain Restrictions
$$q(x)\ne0$$
Horizontal Asymptotes
$$\deg p<\deg q:\ y=0$$ $$\deg p=\deg q:\ y=\frac{\text{lead }p}{\text{lead }q}$$
Slant Asymptote
$$\deg p=\deg q+1$$ $$\text{divide }p(x)\text{ by }q(x)$$
Factor first. Cancel only after recording the original restrictions, because canceled factors become holes.

Classic Examples

Vertical Asymptote and Hole

Find the domain restrictions, hole, and vertical asymptote for $f(x)=\frac{x^2-4x+3}{x^2-5x+6}$.

Solution Steps
  1. Factor the numerator and denominator.
  2. Record every original denominator restriction.
  3. Cancel the common factor while keeping the restrictions.
  4. Evaluate the simplified function at the canceled value $x=3$.
  5. State the hole: $(3,2)$.
  6. State the vertical asymptote from the remaining denominator: $x=2$.
$$f(x)=\frac{(x-1)(x-3)}{(x-2)(x-3)}$$ $$x\ne2,\quad x\ne3$$ $$f(x)=\frac{x-1}{x-2},\quad x\ne2,\ x\ne3$$ $$y=\frac{3-1}{3-2}=2$$ $$\text{hole at }(3,2)$$ $$\text{vertical asymptote: }x=2$$

Horizontal Asymptote

Find the horizontal asymptote of $g(x)=\frac{2x^2+3x+1}{x^2+4}$.

Solution Steps
  1. Find the degree of the numerator.
  2. Find the degree of the denominator.
  3. Because the degrees are equal, use the ratio of leading coefficients and state the asymptote: $y=2$.
$$\deg(2x^2+3x+1)=2$$ $$\deg(x^2+4)=2$$ $$y=\frac{2}{1}=2$$

Slant Asymptote

Find the slant asymptote of $h(x)=\frac{x^2+2x+3}{x+1}$.

Solution Steps
  1. Divide the numerator by the denominator.
  2. State the slant asymptote: $y=x+1$.
$$\frac{x^2+2x+3}{x+1}=x+1+\frac{2}{x+1}$$ $$\text{slant asymptote: }y=x+1$$

Intercepts

Find the intercepts of $r(x)=\frac{x-2}{x+1}$.

Solution Steps
  1. Set the numerator equal to zero for the x-intercept.
  2. Solve for the x-coordinate.
  3. Evaluate the function at $x=0$ for the y-intercept.
  4. State the x-intercept: $(2,0)$.
  5. State the y-intercept: $(0,-2)$.
$$x-2=0$$ $$x=2$$ $$r(0)=\frac{0-2}{0+1}=-2$$ $$\text{x-intercept: }(2,0)$$ $$\text{y-intercept: }(0,-2)$$

Graphing Checklist

Step Action
1 Factor numerator and denominator.
2 Record domain restrictions from the original denominator.
3 Cancel common factors and mark holes.
4 Use remaining denominator zeros for vertical asymptotes.
5 Use degree comparison or division for end behavior.
6 Find intercepts and use a sign chart to place branches.