Q.Write the domain and extent of the function $f(x) = x^3 + 1$.

Symmetry. A curve $f(x,y)=0$ is:

• symmetric about the $y$-axis if $f(x,y)=f(-x,y)$ for all $(x,y)$ on it (i.e. $(x,y)$ on the curve $\Rightarrow(-x,y)$ is too);
• symmetric about the $x$-axis if $f(x,y)=f(x,-y)$ (i.e. $(x,y)$ on it $\Rightarrow(x,-y)$ is too);
• symmetric about the origin if $f(x,y)=f(-x,-y)$ (i.e. $(x,y)$ on it $\Rightarrow(-x,-y)$ is too — equivalently, the curve is unchanged by a $180^\circ$ rotation about the origin).

Asymptotes. An asymptote is a straight line the curve approaches (the gap shrinking to $0$) as the point on the curve runs off to infinity. Three kinds:

• Horizontal asymptote $y=L$: holds if $\displaystyle\lim_{x\to+\infty}f(x)=L$ or $\displaystyle\lim_{x\to-\infty}f(x)=L$ (the two one-sided limits may give different horizontal asymptotes).
• Vertical asymptote $x=a$: holds if $\displaystyle\lim_{x\to a^{-}}f(x)=\pm\infty$ or $\displaystyle\lim_{x\to a^{+}}f(x)=\pm\infty$ — typically where a rational function's denominator vanishes while the numerator does not.
• Slant (oblique) asymptote $y=mx+c$: occurs for a rational function when the numerator's degree is exactly one more than the denominator's. Found by polynomial long division: writing $\dfrac{p(x)}{q(x)}=(\text{quotient})+\dfrac{\text{remainder}}{q(x)}$, the quotient (a linear expression) is the slant asymptote, since the remainder term $\to0$ as $x\to\pm\infty$.

Sketching a curve $y=f(x)$ — the seven-point checklist (used, in this order, throughout Examples 7.69–7.72 and Exercise 7.9 Q2):

1. Domain and range of $f$.
2. Intercepts — set $y=0$ for $x$-intercepts, $x=0$ for the $y$-intercept (where each exists).
3. Critical points — solve $f'(x)=0$ and note where $f'$ fails to exist.
4. Local extrema — classify each critical point (first or second derivative test) and record the extreme value. …