So, the appropriate domain for this situation is all non-negative real numbers, or in interval notation:...

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So, the appropriate domain for this situation is all non-negative real numbers, or in interval notation: [0,00). b. The range of a function refers to the set of all possible output values that the function can produce. In this case, the function H(t) represents the temperature of the hot chocolate. Looking at the equation: H(t) = 70 + 100* (0.819)^t authe The first term, 70, is a constant that represents the base temperature. The second term, 100* (0.819)^t, represents the increase in temperature due to the exponentiation of 0.819 to the power of t. Since 0.819 is between 0 and 1, raising it to larger and larger powers will cause it to approach zero, resulting in diminishing temperature increases. This means that the temperature of the hot chocolate will increase over time but will asymptotically approach a maximum value (which is the base temperature plus the initial increase). As time goes to infinity, the exponential term will approach zero, so the temperature will approach the constant value of 70. However, it will never quite reach 70 if t is finite and positive. So, the appropriate domain for this situation is all non-negative real numbers, or in interval notation: [0,00). b. The range of a function refers to the set of all possible output values that the function can produce. In this case, the function H(t) represents the temperature of the hot chocolate. Looking at the equation: H(t) = 70 + 100* (0.819)^t authe The first term, 70, is a constant that represents the base temperature. The second term, 100* (0.819)^t, represents the increase in temperature due to the exponentiation of 0.819 to the power of t. Since 0.819 is between 0 and 1, raising it to larger and larger powers will cause it to approach zero, resulting in diminishing temperature increases. This means that the temperature of the hot chocolate will increase over time but will asymptotically approach a maximum value (which is the base temperature plus the initial increase). As time goes to infinity, the exponential term will approach zero, so the temperature will approach the constant value of 70. However, it will never quite reach 70 if t is finite and positive.