Precalculus Rational Functions and Asymptotes Worksheet Solutions (PDF)

\(x^{2}-5x+6=(x-2)(x-3)\), so \(x\neq 2\) and \(x\neq 3\).

\((-\infty,2)\cup(2,3)\cup(3,\infty)\)

\(f(x)=\frac{(x-3)(x+3)}{(x-3)(x+2)}\). The factor \(x-3\) cancels: hole at \(x=3\) with \(y=\frac{3+3}{3+2}=\frac65\). The factor \(x+2\) remains: asymptote \(x=-2\).

Hole \(\left(3,\frac65\right)\); vertical asymptote \(x=-2\)

Equal degrees: divide leading coefficients, \(\frac62\).

\(y=3\)

The numerator has degree 1 and the denominator has degree 2.

\(y=0\)

The degree difference is \(4-2=2\), more than 1.

Neither

\(2x-6=0\) gives \(x=3\). \(f(0)=\frac{-6}{4}=-\frac32\).

\(x\)-intercept \((3,0)\); \(y\)-intercept \(\left(0,-\frac32\right)\)

The denominator is 0 at \(x=3\). The degrees are equal with leading coefficients 2 and 1. \(f(0)=0\).

\(x=3\); \(y=2\); intercept \((0,0)\)

Divide with synthetic division, \(c=3\): \(1,-1,-2\to 1,2,4\). The quotient is \(x+2\).

\(y=x+2\)

Cancel \(x-2\): \(f(x)=x+2\). At \(x=2\), \(y=4\).

\((2,4)\)

Cross multiply: \(3(x+2)=2(x-1)\), \(3x+6=2x-2\), so \(x=-8\). Check: \(\frac{3}{-9}=-\frac13\) and \(\frac{2}{-6}=-\frac13\).

\(x=-8\)

Multiply by \(x(x+1)\): \(2(x+1)+3x=2x(x+1)\), so \(5x+2=2x^{2}+2x\) and \(2x^{2}-3x-2=0\). Factor: \((2x+1)(x-2)=0\). Both \(x=-\frac12\) and \(x=2\) keep the denominators nonzero.

\(x=2\) or \(x=-\frac12\)

Multiply by \(x-3\): \(x=3+2(x-3)=2x-3\), so \(x=3\). But \(x=3\) makes the denominators 0.

No solution (3 is extraneous)

Critical values: \(x=1\) (numerator 0, included) and \(x=-2\) (denominator 0, excluded). Test \(x=-3\): \(\frac{-4}{-1}>0\). \(x=0\): \(\frac{-1}{2}< 0\). \(x=2\): \(\frac{1}{4}>0\).

\((-\infty,-2)\cup[1,\infty)\)

The denominator is \(x-2\). Equal degrees with ratio 3: numerator \(3(x+1)\).

\(f(x)=\frac{3(x+1)}{x-2}\)

\(f(2.01)=\frac{1}{0.01}=100\). \(f(1.99)=\frac{1}{-0.01}=-100\).

\(100\) and \(-100\); \(f\to\infty\) as \(x\to2^{+}\), \(f\to-\infty\) as \(x\to2^{-}\)

Cancel \(x-1\): \(\frac{x+3}{x-5}\). The hole is at \(x=1\), \(y=\frac{4}{-4}=-1\). The asymptote is \(x=5\).

Hole \((1,-1)\); asymptote \(x=5\)

\(A(50)=\frac{1100}{50}=22\). The asymptote \(y=12\): as \(n\) grows, the fixed cost is spread out and the average approaches \(\$12\). Solve \(500+12n\le 15n\): \(n\ge\frac{500}{3}\approx 166.7\).

\(\$22\); \(y=12\); \(n=167\) shirts

Equal degrees, ratio \(\frac11\): \(y=1\) (the fraction can never reach 100%). Solve \(2+x=0.5(10+x)\): \(2+x=5+0.5x\), \(0.5x=3\), \(x=6\).

\(y=1\); \(x=6\) gallons

Multiply by \(2x(x-3)\): \(2(x-3)+2x=x(x-3)\), so \(4x-6=x^{2}-3x\) and \(x^{2}-7x+6=0\), giving \(x=6\) or \(x=1\). If \(x=1\), then \(x-3< 0\), which is impossible. So \(x=6\). Check: \(\frac16+\frac13=\frac12\).

Mei \(6\) hours; Jamal \(3\) hours

\(f(x)=\frac{(x-2)(x+2)}{(x-3)(x+2)}\). Cancel \(x+2\): hole at \(x=-2\), \(y=\frac{-4}{-5}=\frac45\). Vertical asymptote \(x=3\). Equal degrees: \(y=1\). \(x\)-intercept: \(2\). \(y\)-intercept: \(f(0)=\frac{-4}{-6}=\frac23\).

Hole \(\left(-2,\frac45\right)\); \(x=3\); \(y=1\); \((2,0)\); \(\left(0,\frac23\right)\)