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Radical Functions

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

Summary: Finding the domain of radical/root functions.

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=2⁢x−5 y 2x5

Solution

x x≥52 x x 52 since 2⁢x−5≥0 2x5 0 , 2⁢x≥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=4⁢x2−164 y 4 4x216

Solution

−∞−2 ∪2∞ 2 2 since 4⁢x2−16≥0 4x216 0 , 4⁢x2≥16 4 x2 16 , x2≥4 x2 4 , (x≤−2) or (x≥2) x 2 x 2

Exercise 4

y=16−25⁢x2 y 16 25 x 2

Solution

−45 45 45 45 since 16−25⁢x2≥0 16 25 x2 0 , -25⁢x2≥-16 -25 x2 -16 , x2≤1625 x2 1625 , (x≥−45) and (x≤45) x 4 5 x 4 5

Exercise 5

y=(x−7)⁢(x+1) y x 7 x 1

Solution

−∞−1 ∪7∞ 1 7 , (x−7)⁢(x+1)≥0 x 7 x 1 0

Exercise 6

y=2⁢x2−7⁢x+3 y 2 x 2 7 x 3

Solution

−∞1/2 ∪3∞ 12 3 , 2⁢x2−7⁢x+3≥0 2 x 2 7 x 3 0 , (2⁢x−1)⁢(x−3)≥0 2 x 1 x 3 0 , (x≤12) or (x≥3) x 1 2 x 3

Exercise 7

y=x⁢x2+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=6⁢x2+8 y 6x28

Solution

−∞∞ , since 6⁢x2+8≥0 6 x 2 8 0 , 6⁢x2≥−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)−6⁢x2 y 86x2

Solution

No solution since (−8)−6⁢x2≥0 8 6 x2 0 , -6⁢x2≥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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