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Domain of a Square Root Function



The domain of a square root function given by y = k √x is determined by the following:
 
  • Range is a real number. The range is ≥ 0 when k ≥ 0, and the range is < 0 when k < 0.

  • The domain is x ≥ 0.
 
Example:
 

Find the domain and range of the following function
1.
y =
   
  Solution :
   
  As stated above, the linear function in a square root should always have its value greater than or equal to zero.

Therefore, the domain of y will be given by x – 1 ≥ 0 or x ≥ 1.

For the range of y, the value of the square root function will always be greater than or equal to zero.

Hence the range of y is ≥ 0.
   
2.
y =
   
  Solution:
   
  Domain: x ≥ 0
   
 
For the range: since   ≥ 0, y ≥ 0 + 5 or y ≥ 5.
   
3.
y =
   
  Solution :
   
  To find the domain, we use the condition that 20-9x ≥0.
   
  Add 9x on both sides: 20 ≥ 9x.

Divide both sides with 9: 20/9 ≥ x.

Hence the domain of y is x ≤ 20/9.
   
  The range will be the same as for
y=
because the square root function will always give a value ≥ 0.
 

Try this problem
 
1.
Find the domain of
 
a.
 x > 9/5
 
b.
 x < 9/5
 
c.
 x ≥ 9/5
 
d.
 x ≤ 9/5
     
  Answer: C
   
  Explanation :
   
 
The given square root function  is
   
 
The square root function is defined for f(x)≥0.
   
  Hence, f(x)= 5x-9>=0.

Add  9 to both the sides: 5x ≥9.
   
  Divide by 9:

x ≥ 9/5
 


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