To find the domain of the function

$f(x) = \frac{2x - 3}{x^2 + 4},$

we need to determine where the function is defined. A rational function is undefined wherever the denominator is zero.

Step 1: Set the denominator equal to zero and solve for $x$.

$x^2 + 4 = 0.$

Step 2: Solve for $x$.

$x^2 = -4.$

Since the square of a real number is always non-negative, $x^2 = -4$ has no real solutions. This means the denominator is never zero for any real number $x$.

Step 3: Conclusion

Because the denominator is never zero, the function is defined for all real values of $x$.

Thus, the domain of the function is:

$(-\infty, \infty).$

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Would you like a more detailed explanation of this process? Let me know if you have any questions!

Here are some related questions for further exploration:

1. What would happen to the domain if the denominator were $x^2 - 4$ instead of $x^2 + 4$?
2. How can you find the domain of a rational function in general?
3. What would the domain be for the function $\frac{1}{x^2 + 4x - 5}$?
4. How do you determine where a function is continuous based on its domain?
5. How does the presence of square roots in the denominator affect the domain of a function?

Tip: Always check the denominator of a rational function to ensure it’s not equal to zero, as that’s where the function is undefined!