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A function is a mathematical concept that relates an input to an output. Think of it as a machine where you input a number and get an output number. Every unique input maps to exactly one output. There can be many types of functions:

• Linear functions, like a straight line in a graph.
• Quadratic functions, which form parabolas.
• Sine and Cosine functions, which are wavy and periodic.

Understanding functions involves determining their domain, which is the set of all possible input values. In mathematical terms, it’s the set of numbers you can plug into a function without causing any issues like division by zero or the square root of a negative number.

Most basic functions have simple domains. For example, a linear function like \(f(x) = 2x + 3\) has the domain of all real numbers because you can input any real number and it will give you a meaningful output.

The sine function, written as \(\sin(x)\), is a periodic function from trigonometry. It creates a wave-like pattern and is fundamental in understanding oscillations and waves. The sine function operates smoothly over the entire set of real numbers, which means its domain is all real numbers, \((−\infty, \infty)\).

• Sine functions repeat every \(2\pi\) radians, showing the same values in regular intervals.
• \(\sin(0) = 0\), \(\sin(\pi/2) = 1\), and \(\sin(\pi) = 0\) are some key values of the function.
• The function is continuous, meaning it has no breaks or holes in its graph.

In the exercise, our function involves the sine function with an exponential term \(e^{-t}\) inside it. The sine function itself has no domain restrictions, but it's always good to check if other parts of the function, like \(e^{-t}\), add any. Here, \(e^{-t}\) works for all real \(t\), so there are no additional restrictions to the input domain.

The square root function \(\sqrt{x}\) is a classic example of a function with domain restrictions. For the square root function to produce real numbers, the input \(x\) must be non-negative. This means \(x\geq 0\).

• Square root functions are not defined for negative numbers; attempting to find the square root of a negative number results in an imaginary number.
• The graph of \(\sqrt{x}\) starts at the origin (0,0) and curves upward.
• This function is increasing—you feed it larger numbers, and the output gets larger.

For the function \(g(t)=\sqrt{1-2^{t}}\), the condition that makes the function's square root usable is \(1-2^t \geq 0\). Solving this gives \(t \leq 0\). That's why, to keep our function defined and working properly, we restrict \(t\) to values from negative infinity up to 0, which gives us the domain \((−\infty, 0]\).