Tuesday, October 21, 2014

College Algebra, Chapter 3, Review Exercises, Section Review Exercises, Problem 78

Suppose that f(x)=x and g(x)=2x4 find fg, gf, ff and gg and their domains

fg=f(g(x))=f(2x4)=2x4gf=g(f(x))=g(x)=2x4ff=f(f(x))=f(x)=x=4xgg=g(g(x))=g(2x4)=22x44=2(x4)24(x4)=2(x4)24x+16=2(x4)4x+18=2(x4)2(2x9)=(x4)2x9


To find the domain of fg, we want...

x4>0Add 4x>4

Thus, the domain of fg is (4,)

To find the domain of gf, we want x40. The denominator is zero when...

x4=0Add 4x=4Square both sidesx=16


Also, the function involves square root that is defined for only positive values. Thus, the domain of gf is [0,16)(16,)

To find the domain of ff, recall that the function with even roots is defined only for positive values of x. Thus, the domain of ff is [0,)

To find the domain of gg, we want 2x90 the denominator is zero when...

2x9=0Add 92x=9Divide 2x=92

Thus, the domain of gg is (,92)(92,)

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