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Pointwise Operations of Functions

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1. If f (x)= 2X - X -3 and g(x)= X+1, find f + g.

2. if f(x) = 3X-4 and g (x) = X+2 , find f + g

3. if f (x)= 3X-4 and g(x)= X+2 find f * g

4. if f(x)= 3X and g (x)= X-5 find (f*g) (X)

5. if f(x) = 5X+2 an g (x) = 3X-4 find (f*g) (x)


4X + 81 - 36X

X - 25
X + 5.

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Solution Summary

Functions are added and multiplied. The solution is detailed and well presented.

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Pointwise Operations and Characteristic Functions

Let U be a set, suppose f, g : U --> R are functions from U to the set of real numbers R, and Let a E R.
Then f + g, fg. af: U --> R are defined by
(f + g)(x) = f(x) + g(x),
(fg)(x) = f(x)g(x)
(af)(x) = a(f(x))
for all x E R.
If a E R by abuse of notation we regard a as the constant function from U to R defined by a(x) = a for all x E U.
Now regard U as a universal set. For a subset A of U the characteristic fundion of A is the function A : U ?> R defined by
1. Let U be a universal set.
a) Let AB C U. Show that XA =XB if and only if A=B.
b) Show that x0= 0 and xu = 1
c) Suppose that f : U --> R is a function such that un f C {0, 1}. Show that f is the characteristic function of some subset of U.

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