Find the domain and range of h(x)=√1-3x

Answer 1

The domain is the set of values that your input variable, x, can take. Since we are looking at h(x) = √(1 - 3x) and we know that we cannot take the square root of a negative number, 1 - 3x must be greater than or equal to 0 (not negative). Algebraically,

1 - 3x ≥ 0

1 ≥ 3x

1/3 ≥ x

The only limitation on x, and thus the domain, is that x ≤ 1/3. We say that the domain is (-∞, 1/3], remembering to put the square bracket on the right because the inequality is not a strict inequality.

Considering our domain, we can think of the range as the values that we can get out of the function. Start on one end of the domain and consider the output as you move through your domain. At x = 1/3, h(x) = h(1/3) = 0. Moving toward -∞, assume x = 0. Then, at x = 0, h(x) = h(0) = √1 = ±1. We can try another input, x = -8/3. At x = -8/3, h(x) = h(-8/3) = √9 = ±3. Because of the two roots for any given input, you can see that the range will consist of all real numbers which is expressed as the interval (-∞, ∞).