how do you find the domain and range of a half circle

Question

I need some help finding the range of a half circle on a graph.

Answer 1

The domain is the set of x values in which a function is defined.  The range is the set of y values in which the function is defined.

 

 

A circle has a centerpoint (h, k) with radius r.   The equation of a circle is then

 

(x - h)² + (y - k)² = r²

 

 

Lets solve for y from this equation.

 

(y - k)² = r² - (x - h)²

 

(y - k)² = r² - (x² - 2xh + h²)

 

(y - k)² = r² - x² + 2xh - h²

 

y - k = √(r² - x2 + 2xh - h²)

 

y = k + √(r² - x² + 2xh - h²)

 

 

We can find the domain and range of this half circle by setting the value under the square-root greater than or equal to zero.

 

Domain:

 

r² - x² + 2xh - h² ≥ 0

 

 

Range:

 

Once you have your domain, evaluate y by plugging in the lowest possible value of x and the highest possible value of x.

 

 

I hope this makes sense to you.

 

 

Note:  Your domain and range also depends on the orientation of the half circle.  You can either have an upper half, lower half, right half, or left half.