Question

Determine the range of each function
a)f(x)=(x^2+3)^2,XER
Answer is f(x)ER, f(x)>9
b)f(x)=1/3+x^4,XER
Answer is f(x)ER, 0<f(x)<1/3

Please explain how they go those two answers

Answer 1

x^2+3 >= 3 for all x
so, f(x) >= 9

3+x^4 >= 3 for all x
1/(3+x^4) <= 1/3 for all x
as x gets huge, 1/x^4 -> 0 so,
0 < f(x) <= 1/3

wolframalpha.com will graph them for you to see

Answer 2

I suppose that "XER" means the domain (all possible values of x) is all real numbers. And the answers there are both...wrong.
Anyway, range is the set of all possible values of y (or f(x)). Let's analyze each problem. :)

a) f(x) = (x^2+3)^2
Note that x^2 can never be negative, because any real number squared is always greater than or equal to zero. Thus the smallest possible value of x^2 is zero, and if we substitute it in f(x),
f(x) = (0^2 + 3)^2
f(x) = 3^2
f(x) = 9, which is the smallest possible value of f(x)
Thus the range is all real numbers greater than or equal to 9, or f(x) >= 9

b) f(x) = 1/3+x^4
x^4 is also (x^2)^2. Then again, x^4 can never be negative. So the smallest possible value of x^4 is zero, and substituting,
f(x) = (1/3) + 0^4
f(x) = 1/3
Thus the range is all real numbers greater than or equal to 1/3, or f(x) >= 1/3

Hope this helps~ :3