Given the circle (x+2)^2 + (y-1)^2 = 25, write an equation for the function f whose graph is in the upper half of this circle. give the domain and range of the function f expressed in interval notation. * show work *

 f(x) = 1 + √(- x² - 4x + 21), D: [-7, 3], R: [1, 6]

Explanation:

(x + 2)² + (y - 1)² = 25   ⇒ center (h, k) = (-2, 1) and radius (r) = 5

Equation of a semicircle is to solve the equation for y.  

(x + 2)² + (y - 1)² = 25

               (y - 1)² = 25 - (x + 2)²

               (y - 1)² = 25 - (x² + 4x + 4)

               (y - 1)² = 25 - x² - 4x - 4

               (y - 1)² = - x² - 4x + 21

              √(y - 1)² = √(- x² - 4x + 21)

                  y - 1   = +/- √(- x² - 4x + 21)

                     y     = 1 +/- √(- x² - 4x + 21)

Since we are looking for the top half, the equation is:

                       y = 1 + √(- x² - 4x + 21)

in function format:

                     f(x) = 1 + √(- x² - 4x + 21)

NOTE: The easiest way to find the domain and range is to graph the circle.  Below is how to find the domain and range algebraically.

Domain: square root must be greater than or equal to zero

- x² - 4x + 21 ≥ 0

 x² + 4x - 21 ≤ 0  divided both sides by -1

(x + 7)(x - 3) = 0   factored to find the zeros

 x + 7 = 0      x - 3 = 0    

     x = -7         x = 3

The domain must be between -7 and 3 inclusively ; -7 ≤ x ≤ 3.

Interval Notation: D: [-7, 3]

Range: find the y-value at one of the zeros (x = -7 or x = 3) and the y-value of the vertex (the axis of symmetry goes through the center, x = -2)

f(x) = 1 + √(- x² - 4x + 21)

f(-7) = 1 + √(- (-7)² - 4(-7) + 21)

       = 1 + √(-49 + 28 + 21)

       = 1 + √(0)

        = 1

f(-2) = 1 + √(- (-2)² - 4(-2) + 21)

       = 1 + √(-4 +8 + 21)

       = 1 + √(25)

       = 1 + 5

       = 6

the range must be between 1 and 6 inclusively ; 1 ≤ y ≤ 6

Interval Notation: R: [1, 6]

what are the options? ?

step-by-step explanation:

there has been genes found in the x chromosome of males that have been linked to [homo/hetero]sexuality. in addition to the reign of the x chromosome, a section of chromosome 8 has also been link to, "alot of ppl are so " these facts were proven true after a 2014 study of a gay male and his strait brother. according to newscientist.com,"chromosome 14 and is mainly active in the thyroid, but also the brain. called tshr, it makes a type of receptor protein that recognises and binds to a hormone that stimulates the thyroid. in this way, the gene plays an important role in controlling thyroid function. the fact that tshr seems to be involved in sexual orientation fits with evidence that thyroid function seems to be linked to sexuality."

in conclusion, "ppl are so gay" due to their genetics.