To determine the domain of the composition of functions \( f(g(x)) \), we first need to find the domain of \( g(x) \) and then see how it affects the domain of \( f(x) \).
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Determine the domain of \( g(x) \):
- The function \( g(x) = 2x - 3 \) is a linear function, which means it is defined for all real numbers.
- Therefore, the domain of \( g(x) \) is \( (-\infty, \infty) \).
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Find the output of \( g(x) \):
- The output of \( g(x) = 2x - 3\) can take any real number value as \( x \) varies over all reals.
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Determine the domain of \( f(x) \):
- The function \( f(x) = x^2 - 1 \) is defined for all real numbers as it is a polynomial.
- Therefore, the domain of \( f(x) \) is also \( (-\infty, \infty) \).
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Determine the domain of the composition \( f(g(x)) \):
- To find \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f(2x - 3) = (2x - 3)^2 - 1 \]
- Since \( g(x) \) is defined for all \( x \), and its output is also in the domain of \( f \), the composition \( f(g(x)) \) can take any real input from \( g(x) \).
Thus, the domain of \( f(g(x)) \) is also \( (-\infty, \infty) \).
In conclusion, the domain of \( f(g(x)) \) is:
\[ \boxed{(-\infty, \infty)} \]
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