To determine the domain of the function $f(x) = 2^x - 3$, we need to analyze where the function is defined.

Step-by-Step Process:

1. Understanding the Function: The function $f(x) = 2^x - 3$ is an exponential function where the base is 2. Exponential functions are generally defined for all real numbers unless there are restrictions on the domain (such as dividing by zero or taking the square root of a negative number, which isn't the case here).

2. Checking for Domain Restrictions:

   • Since $2^x$ is defined for all real numbers $x$, and subtracting 3 from this expression does not introduce any restrictions, the function remains defined for all real numbers.

3. Conclusion: The domain of $f(x) = 2^x - 3$ is all real numbers $\mathbb{R}$.

Domain:

$\text{Domain of } f(x) = 2^x - 3 \text{ is } (-\infty, \infty)$

If you would like to see a graph of the function, the graph would show an exponential curve shifted down by 3 units (compared to the standard exponential curve $2^x$).

Graphing Confirmation:

• Using graphing technology like Desmos or a graphing calculator, you'll see the exponential curve starts from $-3$ on the $y$-axis and increases as $x$ becomes positive, confirming that there are no breaks or undefined points in the function.

Would you like to go over the graph details or have any specific questions?

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Here are five related questions:

1. What is the range of the function $f(x) = 2^x - 3$?
2. How would the graph of $f(x) = 2^x - 3$ compare to $f(x) = 3^x - 3$?
3. What is the horizontal asymptote of the function $f(x) = 2^x - 3$?
4. How would the domain change if the function was $f(x) = \log(x - 3)$?
5. What transformations are applied to the parent function $2^x$ to get $f(x) = 2^x - 3$?

Tip: For exponential functions, pay attention to shifts, as they impact the range and the asymptotes but not the domain.