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# Finding the Domain of 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 x0 x x 0

PRACTICE - Find the Domain of the following:

## Exercise 1

y=2x5 y 2x5

### Solution

x x52 x x 52 since 2x50 2x5 0 , 2x5 2x 5 , x52 x 52

## Exercise 2

y=7x4 y 4 7x

### Solution

x x7 x x 7 since 7x0 7x 0 , x7 x 7 , x7 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=4x2164 y 4 4x216

### Solution

2 2 2 2 since 4x2160 4x216 0 , 4x216 4 x2 16 , x24 x2 4 , (x2)(x2) x 2 x 2

## Exercise 4

y=1625x2 y 16 25 x 2

### Solution

45 45 45 45 since 1625x20 16 25 x2 0 , -25x2-16 -25 x2 -16 , x21625 x2 1625 , (x45)(x45) x 4 5 x 4 5

## Exercise 5

y=(x7)(x+1) y x 7 x 1

### Solution

1 7 1 7 , (x7)(x+1)0 x 7 x 1 0

## Exercise 6

y=2x27x+3 y 2 x 2 7 x 3

### Solution

1/2 3 12 3 , 2x27x+30 2 x 2 7 x 3 0 , (2x1)(x3)0 2 x 1 x 3 0 , (x12)(x3) x 1 2 x 3

## Exercise 7

y=xx2+4 y x x 2 4

### Solution

, since x2+40 x 2 4 0 , x24 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+80 x 8 0 , x8 x 8 , x8 x 8

## Exercise 9

y=6x2+8 y 6x28

### Solution

, since 6x2+80 6 x 2 8 0 , 6x28 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)6x20 8 6 x2 0 , -6x28 -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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