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

1 Answer
Stefan V.
Aug 3, 2015

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.

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