Question

Find the domain of each function.

18. (a) \(g\left( t \right) = \sqrt {{\bf{1}}{{\bf{0}}^t} - {\bf{100}}} \)

(b) \(g\left( t \right) = sin\left( {{e^t} - 1} \right)\)

Short answer

(a) The domain of the function is \(\left\{ {\left. t \right|t \ge 2} \right\}\), or \(\left( {2,\infty } \right)\).

(b) The domain of the function is \(\mathbb{R}\).

Step-by-step solution

The domain of the function

The function \(g\left( t \right) = \sqrt {{{10}^t} - 100} \) is defined if the function \({10^t} - 100\) is 0 or positive.

So,\({10^t} - 100 \ge 0\).

Now, the value of\(t\)for which the function is defined is shown below:

\(\begin{aligned}{10^t} - 100 \ge 0\\{10^t} \ge 100\\{10^t} \ge {10^2}\\t \ge 2\end{aligned}\)

So, the function is defined when\(t \ge 2\).

Thus, the domain of the given function is \(\left\{ {\left. t \right|t \ge 2} \right\}\), or \(\left( {2,\infty } \right)\).

The domain of the function

Consider the function \(g\left( t \right) = \sin \left( {{e^t} - 1} \right)\).

It is known that the domain of sine and exponential function is a real number\(\mathbb{R}\).

Thus, the domain of the given function is \(\mathbb{R}\).