Question

how do you find the range of
f(x) = (x + 5)^2 + 8

Answer 1

I assume, you're looking for an algebraic method , as it's easy in graphs.

Find the inverse of this function:
Let y = f(x)
y = (x+5)^2 + 8
y - 8 = (x+5)^2
sqrt(y - 8) - 5 = x

x = sqrt(y - 8) - 5 y-8 >= 0
y >= 8

]inf, 8]

Answer 2

the range are all the f(x) values that you can get as you change x

it helps to know the range of
f(x)= x^2

do you know that x^2 is 0 or positive, not matter what x is ?

Answer 3

@TrojanPoem that helps a lot. Thank you.

Answer 4

similarly , the smallest (x+5)^2 can be is 0
so the smallest f(x) can be is 0+8= 8
and of course, you can get as big a number as you like if you make x big enough
so [8, +infinity)

Answer 5

Any time.