Lesson 45.

NOTE: Be sure to read  Homework Format  and Homework Writing Policy BEFORE writing up your solutions to be turned in.
PROBLEMS WILL NOT BE GRADED AND NO CREDIT WILL BE GIVEN if these guidelines are not followed.

Last modified Apr 12, 2008 10:40 AM

Due Friday, April 11

All functions considered below are assumed to have domain and codomain equal to the set of real numbers.
Sect. 7:
  1. Consider the function  f  defined as follows:
        For  x ≤ 0,  f(x) = x;
        For  x > 0,  f(x) = x + 1.
    Determine whether  f  is injective, and prove your answer.

  2. Consider the function  f  defined as follows:
        For  x ≤ 0,  f(x) = -x;
        For  x > 0,  f(x) = x + 1.
    Determine whether  f  is injective, and prove your answer.

  3. At least one of the two preceding functions is injective. Take one of them which is injective and construct explicitly a left inverse. Prove everything you need to prove to show that your function is a left inverse (show that the function YOU CONSTRUCT is a left inverse of the given function).

  4. (New April 8.)
    Keep in mind the assumptions on domain and codomain given at the beginning of this Lesson.  Consider the function  exp  defined by  exp(x) = ex  for each real number  x.  Use a graphical argument (usually not allowed in this course) to explain why this function is injective.  Find a left inverse and explain why your alleged left inverse is really a left inverse (in contrast to the preceding problem, a precise formal proof not required). Be sure to check the definition of left inverse to see what the domain and codomain of the left inverse should be.


NOTE: Be sure to read  Homework Format  and Homework Writing Policy BEFORE writing up your solutions to be turned in.
PROBLEMS WILL NOT BE GRADED AND NO CREDIT WILL BE GIVEN if these guidelines are not follows.


NOTE: As stated on the Course Home Page, all due dates are tentative. Assignments, or parts of assignments, may be postponed to a later date.


Last modified Apr 12, 2008 10:40 AM

Go to Lesson 42a (due April 14).

Go to Lesson 46 (due April 14).

Go to Lesson 45a (due April 18).

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