A complete rational-function analysis combines algebra, limits, derivatives, and graph interpretation. Work in a fixed order so that excluded values and asymptotes are not lost.
This guide analyzes
\[f(x)=\frac{x^2+1}{x-1}.\]
1. Determine the domain
The denominator is zero at \(x=1\), so
\[\operatorname{Dom}(f)=(-\infty,1)\cup(1,\infty).\]
No factor cancels with the numerator, so the excluded value produces a vertical asymptote rather than a removable hole.
2. Find intercepts
The y-intercept is
\[f(0)=\frac1{-1}=-1,\]
so the graph passes through \((0,-1)\). For x-intercepts, solve \(x^2+1=0\). There are no real solutions, so the graph does not cross the x-axis.
3. Find asymptotes by division
Polynomial division gives
\[\frac{x^2+1}{x-1}=x+1+\frac{2}{x-1}.\]
Therefore:
- vertical asymptote: \(x=1\);
- oblique asymptote: \(y=x+1\).
The remainder term approaches zero as \(|x|\to\infty\), which confirms the oblique asymptote.
4. Analyze increasing and decreasing intervals
Differentiate the divided form:
\[f'(x)=1-\frac{2}{(x-1)^2}=\frac{x^2-2x-1}{(x-1)^2}.\]
The critical x-values satisfy
\[x^2-2x-1=0\Longrightarrow x=1\pm\sqrt2.\]
Because the denominator of \(f’\) is positive on the domain, the numerator determines the sign:
- increasing on \(( -\infty,1-\sqrt2)\);
- decreasing on \((1-\sqrt2,1)\);
- decreasing on \((1,1+\sqrt2)\);
- increasing on \((1+\sqrt2,\infty)\).
5. Find local extrema
At \(x=1-\sqrt2\), the derivative changes from positive to negative, giving a local maximum:
\[f(1-\sqrt2)=2-2\sqrt2.\]
At \(x=1+\sqrt2\), the derivative changes from negative to positive, giving a local minimum:
\[f(1+\sqrt2)=2+2\sqrt2.\]
6. Analyze concavity
Differentiate again:
\[f”(x)=\frac{4}{(x-1)^3}.\]
- For \(x<1\), \(f”(x)<0\): concave down.
- For \(x>1\), \(f”(x)>0\): concave up.
Concavity changes at \(x=1\), but that value is not in the domain. Therefore the graph has no inflection point.
7. Check one-sided behavior near the vertical asymptote
From \(f(x)=x+1+2/(x-1)\):
\[\lim_{x\to1^-}f(x)=-\infty,\qquad \lim_{x\to1^+}f(x)=+\infty.\]
8. Build the graph and variation table
Enter (x^2+1)/(x-1) in the Graphing Calculator. Add x+1 as a dashed comparison line and mark \(x=1\). Then use the Variation Table tool to verify the derivative signs and extrema.
Checklist for any rational function
- Factor before canceling and record all original excluded values.
- Distinguish holes from vertical asymptotes.
- Find intercepts from the simplified numerator and direct substitution.
- Use degree comparison or division for end behavior.
- Split derivative sign charts at discontinuities as well as critical points.
- Do not label an excluded point as an inflection point.
For focused explanations, review asymptotes, local extrema, and concavity.