What is the domain and range of #y = sqrt(x-10) + 5#?

1 Answer
Aug 3, 2015

Answer:

Domain: #[10, +oo)#
Range: #[5, +oo)#

Explanation:

Let's start with the domain of the function.

The only restriction you have will depend on #sqrt(x-10#. Since the square root of a number will produce a real value only if that number if positive, you need #x# to satisfy the condition

#sqrt(x-10)>=0#

which is equivalent to having

#x-10 >=0 => x>=10#

This means that any value of #x# that is smaller than #10# will be excluded from the function's domain.

As a result, the domain will be #[10, +oo)#.

The range of the function will depend on the minimum value of the square root. Since #x# cannot be smaller than #10#, #f(10# will be the starting point of the function's range.

#f(10) = sqrt(10-10) + 5 = 5#

For any #x>10#, #f(x)>5# because #sqrt(x-10)>0#.

Therefore, the range of the function is #[5, +oo)#

graph{sqrt(x-10) + 5 [-3.53, 24.95, -3.17, 11.07]}

SIDE NOTE Move the focus of the graph 5 points up and 10 points to the right of the origin to see function.