Functions with prescribed properties (asymptotes, inflection, monotonicity)

Text

Write a function (a different one for each item) that:

• (a) is always increasing but is not a line;
• (b) has as oblique asymptote a line parallel to $y=2x$;
• (c) has a vertical asymptote at $x=-3$ and a horizontal asymptote at $y=1$;
• (d) has an inflection point at $x=0$.

Solution

A construction exercise: one valid example is required for each property.

(a) The exponential is strictly increasing everywhere but is not a line: $\boxed{y=e^{x}}$ Indeed $y'=e^{x}>0$ for every $x$, and the graph is not straight. ($y=x^3$ would work too.)

(b) A line parallel to $y=2x$ has slope $2$; just add to $2x$ a term that vanishes at infinity: $\boxed{y=2x+\dfrac{1}{x}}$ As $x\to\pm\infty$ we get $y-2x=\dfrac1x\to0$, so $y=2x$ is an oblique asymptote (slope $2$, parallel to $y=2x$).

(c) Start from a shifted hyperbola: a denominator vanishing at $x=-3$ (vertical asymptote) and limit $1$ at infinity (horizontal asymptote): $\boxed{y=1+\dfrac{1}{x+3}}$ As $x\to-3$ the term $\tfrac{1}{x+3}\to\pm\infty$: vertical asymptote $x=-3$. As $x\to\pm\infty$ we get $y\to1$: horizontal asymptote $y=1$.

(d) The cubic has an inflection exactly at the origin: $\boxed{y=x^{3}}$ Indeed $y''=6x$ vanishes and changes sign at $x=0$: concave down for $x<0$, concave up for $x>0$, hence an inflection at $x=0$.