How do I find the domain and range of: 𝑦 = 2[−3(𝑥 + 1)]^2 + 4 without graphing. And is it a function?

𝑦 = 2[−3(𝑥 + 1)]² + 4

// Look for restrictions for the x-values; for this equation,

// there aren't any restrictions: x can be negative, zero, or

// positive, so

DOMAIN = (-∞,+∞)

// Now look for restrictions for the corresponding y-values;

// y will always be positive because [-3(x+1)]² is always

// positive.

// if x < -1, y is positive and greater than 4

// if x = -1, y =  2[−3(-1 + 1)]² + 4 = 2[0]+4 = 4 so the

// smallest value for y will be 4.

// if x > -1, y is positive and greater than 4

RANGE = [4,+∞)

YES, 𝑦 = 2[−3(𝑥 + 1)]² + 4 is a FUNCTION

To answer your question with regards to the effects of the leading 2:

The 2 makes the graph NARROWER by a factor of 2.

If the number had been 1/2, the graph would have been WIDER.

Refer to the bottom graph to see how transitions effect the base graph y = x²

For verification, see graph below.

𝑦 = 2[−3(𝑥 + 1)]^2 + 4 is NOT an absolute value function 

𝑦 = 2[−3(𝑥 + 1)]^2 + 4  = 18(x+1)^2 + 4

Domain = R or (-∞ , ∞)

Range = [4 , ∞)