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:

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.

Find the domain of

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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*}\]