Problem Set - Graphing Other Functions

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 fx=x.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=x.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=x-5.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=x2-1.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=1,if x>0,0,if x=0,-1,if x<0.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=x-11/3.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=x2/3.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=1x-12/3.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=x1-x2/5.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=x2/36-x1/3.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=y, where x+y=1.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=y, where x2/3+y2/3=a2/3, where a is a constant.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=y, where x=acos3θ and y=asin3θ.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=sinx.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=sin2x-x.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=sinx-cosx, for -π/3<x<0.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=2cosx-sin2x.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=sinxx.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=sin1/x.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=e-x.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=e1/x.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=e-x2.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx has asymptotes).
Let fx=ln4-x2.
  1. Graph fx.
  2. Determine where fx is defined.
  3. Determine where fx is continuous.
  4. Determine where fx is differentiable.
  5. Determine where fx is increasing and where it is decreasing.
  6. Determine where fx is concave up and where it is concave down.
  7. Determine what the critical pionts of fx are.
  8. Determine what the points of inflection of fx are.
  9. Determine what the asymptotes to fx are (if fx 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)