the expression inside the square root sign has to be >= 0.
the denominator can't be equal to 0.
let's take a look at the square root first.
sqrt(x+4) will be >= 0 when x is >= -4.
so the value of x has to be greater than ot equal to -4.
that's one of the requirements of the domain.
let's take a look at the denominator next.
the denominator will be 0 when x is equal to -4 because sqrt(-4+4) is equal to sqrt(0) which is equal to 0.
the value of x can't be equal to -4.
combine must be >= -4 and can't be -4 and your domain is all real values of x such that x > -4.
what happens to your range when x > -4?
it will always be positive because the denominator will always be positive.
it cannot be 0 becauswe the numerator will always be equal to 1.
it can grow to infinity so there is no upper bound to its value.
you have:
domain is all values of x > -4.
range is all values of y > 0.
her's a graph of the function.
hard to see, but some select data points will show you what i mean.
when x = -3.9999999999, 1/sqrt(x+4) will be equal to 1/sqrt(.00000000001) which is equal to 100,000 or thereabouts.
that's a pretty high number and will get even higher as x approaches even closer to -4.
x can't be equal to -4 because then the denominator will be equal to 0 which is a nono.
as x approaches infinity, the denominotr will approach infinity and so the numerator will approach 0.
for example, when x = 10000000, then y = 1 / sqrt(1000000004) which is equal to 3.16 * 10^-5 or thereabouts.
that's a small number very close to 0 which gets even closer to 0 as x gets even larger.
