1. In high school, you learn that a function f : R R is one-to-one if its graph passes
the horizontal line test. Explain what this means and why it agrees with our definition
of a one-to-one function f : R R.
Consider the function sin : R R. Restrict the domain and target so that the
function is a bijection. Use this to determine the domain and range of the inverse trigono-
metric function aresin. Do the same for cosine.
3. Consider the function f(x) = 3r-6
(a) Explicitly find a function g(y) that is inverse to f. You will do this by solving for I
in the equation y = 32-6
(b) The functions f and g are inverse functions on what sets? (i.e. What is domain and
range of f(x) and g(y)?)
(c) Show explicitly (algebraically) that (gof)(z) = x and (fog)(y) = y for I E Domain(f)
and y € Domain(g).
4. Consider the function
(a) Try to find an inverse to f- explicitly, using algebra. (You will probably be unsuc-
(b) Use calculus to show that f R - (0. 00) is a bijection.
5. Suppose f : A - B and g : B C are both onto functions. Show that (gof):A+C
6. Suppose that f : A -> B and g : B - C are invertible functions. Show that
(g 0 f) : A - C is invertible, and
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