What is the domain and range of #r(x)= -3sqrt(x-4) +3#?

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Answer 1 · 2015-08-21T14:09:52

Domain: #[4, +oo)#
Range: #(-oo, 3]#

Your function is defined for any value of #x# that will not make the expression under the square root negative.

In other words, you need to have

#x-4>=0 implies x>=4#

The domain of the function will thus be #[4, +oo)#.

The expression under the square root will have a minimum value at #x = 4#, which corresponds to maximum value of the function

#r = -3 * sqrt(4-4) + 3#

#r = -3 * 0 + 3#

#r = 3#

For any value of #x>4#, you have #x-4>0# and

#r = underbrace(-3 * sqrt(x-4))_(color(blue)(<-3)) + 3 implies r < 3#

The range of the function will thus be #(-oo, 3]#.

graph{-3 * sqrt(x-4) + 3 [-10, 10, -5, 5]}