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Finding the Domain of Simple Rational Functions

Module by: Pradnya Bhawalkar, Kim Johnston

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

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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

{x|x≠0}xx0

Exercise 1

Find the domain of y=1x−5 y1x5

Solution

{x|x≠5}xx5 since x−5≠0x50, x≠5x5

Exercise 2

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

Solution

{x|x≠7}xx7 since x−7≠0x70, x≠7x7

Exercise 3

Find the domain of y=7x5−2x y7x52x

Solution

{x|x≠52}xx52 since 5−2x≠052x0, x≠52x52

Exercise 4

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

Solution

{x|x≠3∨-7} x x 3 7 since x≠3 x 3 and x≠-7 x 7

Exercise 5

Find the domain of y=7x2x2−7x+3 y 7 x 2 x 2 7 x 3

Solution

{x|x≠12∨3} x x 12 3 since 2x2−7x+3≠0 2 x 2 7 x 3 0 , 2x−1x−3≠0 2x 1 x 3 0 , 2x−1≠0 2x 1 0 and x−3≠0 x 3 0 , x≠12 x 12 and x≠3 x 3

Exercise 6

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

Solution

{x|x≠-5} x x 5 since x+52≠0 x5 2 0 , x+5≠0 x5 0 , x≠-5 x 5

Exercise 7

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

Solution

{x|x∈ℝ} x x since x2+25≠0 x 2 25 0 , x2≠-25 x2 25 , x∈ℝ x

Exercise 8

Find the domain of y=x−7x2+2 y x 7 x 2 2 -1

Solution

{x|x∈ℝ} x x since x2+2≠0 x 2 2 0 , x2≠-2 x2 2 , x∈ℝ x

Exercise 9

Find the domain of y=5|x−3| y 5 x3

Solution

{x|x≠3}xx3 since |x−3|≠0 x3 0 , x−3≠0 x3 0 , x≠3 x 3

Exercise 10

Find the domain of y=4|x|−4 y 4 x 4

Solution

{x|x≠-4∨4} x x 4 4 since |x|−4≠0 x 4 0 , |x|≠4 x 4 , x≠-4 x -4 and x≠4 x 4

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