How do you find the domain and range of # y = -sqrt(x+6)+7 #?

Question

Hazel Anglin · Accepted Answer

The domain is #x in [-6,+oo]#. The range is #y in (-oo,7]#

The role is

#y=-sqrt(x+6)+7#

What's under the square root sign must be #>=0#

Consequently,

#x+6>=0#

#=>#, #x>=-6#

The domain is #x in [-6,+oo]#

When #x=-6#

#y=-sqrt(-6+6)+7=0+7=7#

And when #x-> +oo#

#y->(-sqrt(+oo+6)+7)=-oo#

The range is #y in (-oo,7]#

graph{sqrt(x+6)+7 [-28.34, 23.65, 71.9, -32.17]}

Eva Adamson · Answer

undefined To find the domain of the function $ y = -\sqrt{x + 6} + 7 $, we need to determine the values of $ x $ for which the function is defined. Since we cannot take the square root of a negative number in the real number system, the expression under the square root, $ x + 6 $, must be greater than or equal to 0:

$$ x + 6 \geq 0 $$

Solving for $ x $, we have:

$$ x \geq -6 $$

So, the domain of the function is $ x \geq -6 $.

To find the range of the function, we need to consider the behavior of the function as $ x $ varies. Since the square root function is always non-negative, $ -\sqrt{x + 6} $ will always be less than or equal to 0. Adding 7 to this result will yield values ranging from 7 down to 7 minus the maximum value of $ -\sqrt{x + 6} $. As $ x $ increases without bound, $ -\sqrt{x + 6} $ decreases without bound. Thus, the range of the function is:

$$ y \leq 7 $$

Therefore, the domain of the function is $ x \geq -6 $ and the range is $ y \leq 7 $.