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Finding the Domain of Simple Rational Functions

Module by: Pradnya Bhawalkar, Kim Johnston. E-mail the authors

Summary: Finding the domain of rational functions with no other complications such as roots/radicals.

For fractions, the denominator (the bottom) of the fraction cannot equal 00. Determine domain restrictions by setting the denominator equal to 00 and solving.

Example 1

Find the domain of y=1x y1x

xx0xx0

Exercise 1

Find the domain of y=1x5 y1x5

Solution

xx5xx5 since x50x50, x5x5

Exercise 2

Find the domain of y=4x+3x7 y 4 x 3 x 7

Solution

xx7xx7 since x70x70, x7x7

Exercise 3

Find the domain of y=7x52x y7x52x

Solution

xx52xx52 since 52x052x0, x52x52

Exercise 4

Find the domain of y=2(x3)(x+7) y 2 x 3 x 7

Solution

x x37 x x 3 7 since x3 x 3 and x7 x 7

Exercise 5

Find the domain of y=7x2x27x+3 y 7 x 2 x 2 7 x 3

Solution

x x123 x x 12 3 since 2x27x+30 2 x 2 7 x 3 0 , (2x1)(x3)0 2x 1 x 3 0 , 2x10 2x 1 0 and x30 x 3 0 , x12 x 12 and x3 x 3

Exercise 6

y=2x+1x+52-1 y 2 x 1 x 5 2 -1

Solution

x x5 x x 5 since x+520 x5 2 0 , x+50 x5 0 , x5 x 5

Exercise 7

Find the domain of y=x+3x2+25-1 y x 3 x 2 25 -1

Solution

x xR x x since x2+250 x 2 25 0 , x225 x2 25 , xR x

Exercise 8

Find the domain of y=x7x2+2-1 y x 7 x 2 2 -1

Solution

x xR x x since x2+20 x 2 2 0 , x22 x2 2 , xR x

Exercise 9

Find the domain of y=5|x3| y 5 x3

Solution

xx3xx3 since |x3|0 x3 0 , x30 x3 0 , x3 x 3

Exercise 10

Find the domain of y=4|x|4 y 4 x 4

Solution

x x44 x x 4 4 since |x|40 x 4 0 , |x|4 x 4 , x-4 x -4 and x4 x 4

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