Let's determine the domain of the composite function [tex]\((f \circ g)(x)\)[/tex], which is defined as [tex]\(f(g(x))\)[/tex].
### Step-by-Step Solution:
1. Understand the individual functions:
- Function [tex]\(f(x)\)[/tex] is defined as [tex]\(f(x) = x^2 - 1\)[/tex].
- Function [tex]\(g(x)\)[/tex] is defined as [tex]\(g(x) = 2x - 3\)[/tex].
2. Determine the domain of [tex]\(g(x)\)[/tex]:
- [tex]\(g(x) = 2x - 3\)[/tex] is a linear function.
- Linear functions are defined for all real numbers.
- Hence, the domain of [tex]\(g(x)\)[/tex] is [tex]\((-\infty, \infty)\)[/tex].
3. Determine the domain of [tex]\(f(x)\)[/tex]:
- [tex]\(f(x) = x^2 - 1\)[/tex] is a polynomial function.
- Polynomial functions are also defined for all real numbers.
- Thus, the domain of [tex]\(f(x)\)[/tex] is [tex]\((-\infty, \infty)\)[/tex].
4. Composite function [tex]\(f(g(x))\)[/tex]:
- The domain of the composite function [tex]\(f(g(x))\)[/tex] is determined by the domain of [tex]\(g(x)\)[/tex] first and then the domain of [tex]\(f(x)\)[/tex] applied to [tex]\(g(x)\)[/tex].
5. Apply the domains:
- Since both [tex]\(g(x)\)[/tex] and [tex]\(f(x)\)[/tex] are defined for all real numbers, the composite function [tex]\(f(g(x))\)[/tex] will also be defined for all real numbers.
6. Conclusion:
- The domain of [tex]\((f \circ g)(x)\)[/tex] is [tex]\((-\infty, \infty)\)[/tex].
Therefore, the domain of [tex]\((f \circ g)(x)\)[/tex] is [tex]\((-\infty, \infty)\)[/tex]. The correct answer is:
[tex]\[
\boxed{(-\infty, \infty)}
\][/tex]
