How to find the domain of each function in interval notation

These are questions from my summer math assignment that I am having problems with. I have looked around online and in notes and can't figure out how to find the domain of each of these functions. Please show work and explain how you got your answer and put them in interval notation.

f(x)=x^2-4x+7
f(x)=(x-2)/(x+4)
f(x)=(x^2+2x)/(x+1)
f(x)=(x+4)/(x^2-4)
f(x)=(x+6)/(x^2+5)
f(x)=x/(x^3+8)
f(x)=(x+3)/the square root of (x^2-9)
f(x)=the square root of (x+7)/(x^2+7)

f(x)=x^2-4x+7

DOMAIN: ℝ or (-∞,∞)

f(x)=(x-2)/(x+4)

DOMAIN: ℝ \ {-4} or (-∞,-4) U (-4,∞)

f(x)=(x^2+2x)/(x+1)

DOMAIN: ℝ \ {-1} or (-∞,-1) U (-1,∞)

f(x)=(x+4)/(x^2-4)

DOMAIN: ℝ \ {-2,2} or (-∞,-2) U (-2,2) U (2,∞)

f(x)=(x+6)/(x^2+5)

DOMAIN: ℝ or (-∞,∞)

f(x)=x/(x^3+8)

DOMAIN: ℝ \ {2} or (-∞,2) U (2,∞)

f(x)=(x+3)/[sqrt(x^2-9)]

DOMAIN: (-∞,-3] U [3,∞)

f(x)=sqrt[(x+7)]/(x^2+7)

DOMAIN: [-7,∞)

To solve these, pay attention to the denominator and make sure it doesn't equal zero. Also, be attentive for square roots, since there is no real root of a negative number.

f(x)=x^2-4x+7 domain (-∞,∞)
f(x)=(x-2)/(x+4) domain (-∞,-4) U (-4,∞) or x≠±4
f(x)=(x²+2x)/(x+1) domain (-∞,-1)U(-1,∞) or x≠±1
f(x)=(x+4)/(x²-4) domain (-∞,-2)U(-2,2)U(2,∞) or x≠±2
f(x)=(x+6)/(x²+5) domain (-∞,-5)U(-5,∞) or x≠ -5
f(x)=x/(x³+8) domain (-∞,-2)U(-2,∞) or x≠ -2
f(x)=(x+3)/√(x²-9) domain (-∞,-3)U(-3,3)U(3,∞) or x≠±3
f(x)=√(x+7)/(x²+7) domain [-7,∞) or x≥-7