Problem Set - Graphing Other Functions

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 7 December 2009

Graphing Other Functions

	Let $f\left(x\right)=⌊x⌋$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let $f\left(x\right)=\left|x\right|$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let $f\left(x\right)=\left|x-5\right|$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=\left|x^2-1\right|}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=\left\{\begin{array}{ll}1\text{,}& \text{if }x>0\text{,}\\ 0\text{,}& \text{if }x=0\text{,}\\ -1\text{,}& \text{if }x<0\text{.}\end{array}\right.}$ Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)={\left(x-1\right)}^{1/3}}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=x^{2/3}}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=\frac{1}{{\left(x-1\right)}^{2/3}}}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=x{\left(1-x\right)}^{2/5}}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=x^{2/3}{\left(6-x\right)}^{1/3}}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=y}$, where $\sqrt{x}+\sqrt{y}=1$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=y}$, where $x^{2/3}+y^{2/3}=a^{2/3}$, where $a$ is a constant. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=y}$, where $x=a{\cos }^3\theta$ and $y=a{\sin }^3\theta$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=\sin x}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=\sin 2x-x}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=\sin x-\cos x}$, for $-\pi /3<x<0$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=2\cos x-\sin 2x}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=\frac{\sin x}{x}}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=\sin \left(1/x\right)}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=e^{-x}}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=e^{1/x}}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=e^{-x^2}}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=\ln \left(4-x^2\right)}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)