Discussion Forum 1  # What is the domain of (fg)(x)?

More formally:

#g sube A xx B :#

#AA a in A AA b_1, b_2 in B#

#((a, b_1) in g ^^ (a, b_2) in g) => b_1 = b_2#

Use the notation #2^A# to represent the set of subsets of #A# and #2^B# the set of subsets of #B#.

Then we can define the pre-image function:

#bar(g)^(-1): 2^B -> 2^A# by #bar(g)^(-1)(B_1) = {a in A : g(a) in B_1}#

Then the domain of #g# is simply #bar(g)^(-1)(B)#

If #f# is a function that maps some elements of set #B# to elements of a set #C#, then:


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The domain of the (f ·g)(x) consists of all x-values that are in the domain of both f and g. In this example, f has domain {x | x ≠ 0}, and g has domain all real numbers, therefore (f · g)(x) has domain {x | x ≠ 0}, because these values of x are in the domain of both f and g.


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Find the domain of

\

Solution

The domain of $$g(x)$$ consists of all real numbers except $$x=\frac{2}{3}$$, since that input value would cause us to divide by 0. Likewise, the domain of $$f$$ consists of all real numbers except 1. So we need to exclude from the domain of $$g(x)$$ that value of $$x$$ for which $$g(x)=1$$.

\ 4 &=3x-2 \\ 6&=3x \\ x&= 2 \end{align*}\]


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