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Finding the Domain of Radical Functions

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

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Summary: Finding the domain of radical/root functions.

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When finding the domain of even-degree roots, the expression under the radical must be greater than or equal to 0.

Example 1

Find the domain of y=x y x

{x|x≥0} x x 0

PRACTICE - Find the Domain of the following:

Exercise 1

y=2x−5 y 2x5

Solution

{x|x≥52} x x 52 since 2x−5≥0 2x5 0 , 2x≥5 2x 5 , x≥52 x 52

Exercise 2

y=7−x4 y 4 7x

Solution

{x|x≤7} x x 7 since 7−x≥0 7x 0 , -x≥-7 x 7 , x≤7 x 7

The rest of the answers will be expressed in interval notation since that is a simpler way to express answers.

Exercise 3

y=4x2−164 y 4 4x216

Solution

-∞-2⋃2∞ 2 2 since 4x2−16≥0 4x216 0 , 4x2≥16 4 x2 16 , x2≥4 x2 4 , x≤-2∨x≥2 x 2 x 2

Exercise 4

y=16−25x2 y 16 25 x 2

Solution

-4545 45 45 since 16−25x2≥0 16 25 x2 0 , -25x2≥-16 -25 x2 -16 , x2≤1625 x2 1625 , x≥-45∧x≤45 x 4 5 x 4 5

Exercise 5

y=x−7x+1 y x 7 x 1

Solution

-∞-1⋃7∞ 1 7 , x−7x+1≥0 x 7 x 1 0

Exercise 6

y=2x2−7x+3 y 2 x 2 7 x 3

Solution

-∞1/2⋃3∞ 12 3 , 2x2−7x+3≥0 2 x 2 7 x 3 0 , 2x−1x−3≥0 2 x 1 x 3 0 , x≤12∨x≥3 x 1 2 x 3

Exercise 7

y=xx2+4 y x x 2 4

Solution

-∞∞ , since x2+4≥0 x 2 4 0 , x2≥-4 x 2 4 This will always be true, for all real numbers, any number squared is always positive

Exercise 8

y=x+-x+8 y x x8

Solution

-∞8 8 since -x+8≥0 x 8 0 , -x≥-8 x 8 , x≤8 x 8

Exercise 9

y=6x2+8 y 6x28

Solution

-∞∞ , since 6x2+8≥0 6 x 2 8 0 , 6x2≥-8 6 x 2 8 , x2≥-86 x 2 -86 This will always be true, for all real numbers, any number squared is always positive

Exercise 10

y=-8−6x2 y 86x2

Solution

No solution since -8−6x2≥0 8 6 x2 0 , -6x2≥8 -6 x2 8 , x2≥-86 x2 -86 This will never be true, so there is no solution, since any number squared is always positive, so it will never be less than 0.

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