Given a single-variable function $f(x)$, work through these in order. Each step narrows down what the graph can look like before you draw a single curve.

1. Domain

Find every real $x$ for which $f(x)$ is defined: no division by zero, no negative argument under an even root, and so on. Write it as intervals, e.g. $D = (-\infty, 1) \cup (1, \infty)$.

2. Image (range)

Determine the set of values $f(x)$ can take as $x$ runs over the domain, such as $[0, \infty)$ or all of $\mathbb{R}$.

3. Roots (zeros)

Solve $f(x) = 0$. These are the points where the graph meets the $x$-axis.

4. Poles (vertical asymptotes)

Look where a denominator is zero or the function becomes undefined. If $\lim_{x \to c} |f(x)| = \infty$, then $x = c$ is a vertical asymptote.

5. Limits and horizontal or oblique asymptotes

Compute $\lim_{x \to \pm\infty} f(x)$.

• A finite limit $L$ gives a horizontal asymptote $y = L$.
• If $f(x)$ behaves like $mx + b$ for large $|x|$, that line is an oblique asymptote.

Also check the limits at the domain boundaries and any excluded points.

6. Symmetry

• Even function: $f(-x) = f(x)$, symmetric about the $y$-axis.
• Odd function: $f(-x) = -f(x)$, symmetric about the origin.
• Check for periodicity if it is relevant.

7. Extrema (local min / max)

Compute $f'(x)$, then solve $f'(x) = 0$ and note where $f'$ is undefined; these are the critical points. Classify each with the sign change of $f'$ or with the second derivative.

8. Inflection points

Compute $f''(x)$, solve $f''(x) = 0$ (or note where it fails to exist), and confirm that the concavity actually changes there.

9. Final graph

Mark the domain, intercepts, poles, and asymptotes. Use the sign of $f'(x)$ for the intervals of increase and decrease, mark the extrema and inflection points, then sketch the curve to fit everything you found.