Saturday, June 16, 2018

College Algebra, Chapter 3, 3.6, Section 3.6, Problem 42

Determine the functions fg,gf,ff and gg and their domains if f(x)=1x and g(x)=x24x
For fg,

fg=f(g(x))Definition of fgfg=1x24xDefinition of f

Since the function involves square root in the denominator, we want

x24x>0x(x4)>0

The factors on the left hand side are x and x4. These factors are zero when x is and 4, respectively. These numbers divide the number line into interval
(,0),(0,4),(4,)
By testing some points on the interval...



Thus, the domain where x(x4)>0 is (,)(4,)

For gf,

gf=g(g(x))Definition of gfgf=(1x)24(1x)Definition of fgf=1x4xDefinition of g

Since the function is a rational function that invovles square root, so the domain of gf is (0,)

For ff,

ff=f(f(x))Definition of ffff=11xDefinition of fff=11xSimplifyff=4x

We know that if the index is any even number, the radicand can't have a negative value. So the domain if ff is [0,)

For gg,

gg=g(g(x))Definition of gggg=(x24x)24(x24x)Definition of ggg=x48x3+16x24x2+16xSimplifygg=x48x3+12x2+16xDefinition of g

The domain of gg is (,)

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