Final answer:
The domain of g(t) is t ≤ -1 and t ≥ 3. The range of g(t) is g(t) ≥ 0. The values of t for which g(t) is defined are t ≤ -1 and t ≥ 3. To find the values of t when g(t) is equal to 0, we set the expression inside the square root equal to 0 and solve for t.
Step-by-step explanation:
(a) Domain of g(t):
The domain of a function includes all the possible values that the input variable, in this case, t, can take. The expression inside the square root, 3-t² - 2t, must be greater than or equal to 0 since we cannot take the square root of a negative number.
Solving the inequality: 3-t² - 2t ≥ 0, we find the critical points to be t = -1 and t = 3.
Hence, the domain of g(t) is t ≤ -1 and t ≥ 3.
(b) Range of g(t):
The range of a function represents all the possible values that the output variable, g(t), can have. Since the square root function always returns a non-negative value, the range of g(t) will always be greater than or equal to 0.
Hence, the range of g(t) is g(t) ≥ 0.
(c) Values of t for which g(t) is defined:
Since the expression inside the square root must be greater than or equal to 0, we can find the values of t that satisfy this condition.
Solving the inequality: 3-t² - 2t ≥ 0, we find the values of t to be: t ≤ -1 and t ≥ 3.
Therefore, for any value of t that falls within this range, g(t) will be defined.
(d) Solving for t when g(t) = 0:
To find the values of t when g(t) is equal to 0, we set the expression inside the square root equal to 0 and solve for t.
√(3-t² - 2t) = 0
3-t² - 2t = 0
t² + 2t - 3 = 0
Factoring the quadratic equation, we get (t+3)(t-1) = 0
So, t = -3 or t = 1.