Complete Function Analysis (Curve Sketching)

Step 1 — Determining the domain: find the set where $f$ is defined. Exclude:

• Denominators = 0 (the function is undefined where the denominator vanishes)
• Logarithm arguments $\leq 0$ (logarithm defined only for $>0$)
• Even-root arguments $< 0$ (even roots defined only for $\geq 0$)
• Arguments of $\arcsin$ and $\arccos$ outside $[-1,1]$
• Bases of irrational powers (if the base is negative, the power may be undefined)

Tip: solve the most restrictive condition first, then intersect with the others.

Step 2 — Symmetries: check whether $f$ has symmetry properties that simplify the study:

• Even if $f(-x)=f(x)$: graph symmetric about the $y$-axis. Study only $x\geq0$ then reflect.
• Odd if $f(-x)=-f(x)$: graph symmetric about the origin. Study only $x\geq0$ then rotate $180^\circ$.
• Periodic with period $T$: study one interval of length $T$ then replicate.

How to check: compute $f(-x)$ and compare with $f(x)$ and $-f(x)$. Note: the domain must be symmetric; otherwise the function can be neither even nor odd.

Step 3 — Axis intercepts:

• $y$-axis: compute $f(0)$ (if $0\in\text{Dom}$). Gives the point $(0,f(0))$ where the graph meets the vertical axis.
• $x$-axis (zeros): solve $f(x)=0$. The solutions are points where the graph touches or crosses the horizontal axis. Often it suffices to know the number and approximate position of zeros.

Step 4 — Sign of $f(x)$: determine the intervals where $f(x)>0$ (graph above $x$-axis) and $f(x)<0$ (graph below $x$-axis).

Sign chart method: factor $f$ (numerator and denominator if rational), study the sign of each factor separately, and combine them using the rule of signs ($+\cdot+ = +$, $+\cdot- = -$, etc.).

Key rules: the sign of $f$ changes only at points where:

• $f$ vanishes (zeros of $f$)
• $f$ is undefined (discontinuities, vertical asymptotes)

Between these points, the sign remains constant (verify with a test point).

Compute limits at domain boundaries: accumulation points excluded from the domain and $\pm\infty$.

Vertical asymptote at $x=x_0$: occurs when $\lim_{x\to x_0^\pm}|f(x)|=+\infty$. Typically where the denominator vanishes but the numerator does not. Distinguish behaviour from left and right ($+\infty$ or $-\infty$).

Horizontal asymptote $y=L$: $\lim_{x\to\pm\infty}f(x)=L$ with $L$ finite. The curve approaches the line $y=L$ for large $|x|$.

Oblique (slant) asymptote $y=mx+q$: when the limit at infinity is not finite but the function grows linearly:

1. $m = \displaystyle\lim_{x\to\pm\infty}\frac{f(x)}{x}$ (must exist, be finite, $m\neq0$)
2. $q = \displaystyle\lim_{x\to\pm\infty}[f(x)-mx]$ (must exist, be finite)

Note: horizontal and oblique asymptotes cannot coexist on the same side. If $m=0$ we have a horizontal asymptote, if $m\neq0$ we have an oblique one.

Compute $f'(x)$ and study its sign to determine the monotonicity of the function.

Monotonicity rule (consequence of Lagrange's theorem):

• $f'(x)>0$ on $(a,b)$ $\Rightarrow$ $f$ is strictly increasing on $[a,b]$
• $f'(x)<0$ on $(a,b)$ $\Rightarrow$ $f$ is strictly decreasing on $[a,b]$
• $f'(x)=0$ on $(a,b)$ $\Rightarrow$ $f$ is constant on $[a,b]$

Critical points: points $x_0$ where $f'(x_0)=0$ (horizontal tangent) or $f'(x_0)$ does not exist. Not all critical points are local extrema!

First derivative test (sign change of $f'$):

• $f'$ changes $+\to-$ at $x_0$ $\Rightarrow$ local maximum
• $f'$ changes $-\to+$ at $x_0$ $\Rightarrow$ local minimum
• $f'$ does not change sign $\Rightarrow$ horizontal inflection point (not an extremum)

Second derivative test (faster when $f''$ is easy to compute):

• $f'(x_0)=0$ and $f''(x_0)>0$ $\Rightarrow$ local minimum
• $f'(x_0)=0$ and $f''(x_0)<0$ $\Rightarrow$ local maximum
• $f'(x_0)=0$ and $f''(x_0)=0$ $\Rightarrow$ inconclusive (use first derivative test)

Compute $f''(x)$ and study its sign to determine concavity.

Concavity:

• $f''(x)>0$ on $(a,b)$ $\Rightarrow$ $f$ is convex (concave up, $\cup$): the curve lies above its tangents, like $y=x^2$.
• $f''(x)<0$ on $(a,b)$ $\Rightarrow$ $f$ is concave (concave down, $\cap$): the curve lies below its tangents, like $y=-x^2$.

Inflection points: points where concavity changes sign. At an inflection, the curve crosses its tangent.

• Necessary condition: $f''(x_0)=0$ or $f''(x_0)$ does not exist
• Sufficient condition: $f''$ changes sign at $x_0$ (from $+$ to $-$ or vice versa)
• Warning: $f''(x_0)=0$ is NOT sufficient — always verify the sign change! Counterexample: $f(x)=x^4$ at $x=0$ has $f''(0)=0$ but it is not an inflection.

Horizontal inflection: $f'(x_0)=0$ and $f''(x_0)=0$ with concavity change — the tangent is horizontal but it is neither a maximum nor a minimum (e.g., $f(x)=x^3$ at $x=0$).

Step 8 — Summary table: collect all notable points in increasing order of $x$ and build a table summarising the function's behaviour.

Standard columns: $x$ (notable points and intervals), $f'$ (sign), $f$ (trend: $\nearrow$ increasing, $\searrow$ decreasing, $M$ max, $m$ min, $\cup$ convex, $\cap$ concave), $f''$ (sign).

Step 9 — Qualitative graph:

1. Draw asymptotes (dashed lines)
2. Mark computed notable points (axis intercepts, maxima, minima, inflections)
3. Sketch the curve respecting: sign (above/below $x$-axis), monotonicity, concavity, asymptotic behaviour

Final checks:

• Does the curve approach the asymptotes without crossing them?
• Are maxima and minima genuine turning points of monotonicity?
• Is the concavity correct in each interval?
• Is the graph consistent with all collected information?