To find the domain of a composition of functions $(f \circ g)(x) = f(g(x))$, follow these simple steps:

Steps to Find the Domain of a Composition of Functions:

1. Find the domain of the inner function $g(x)$:

   • The domain of $g(x)$ consists of all $x$-values for which $g(x)$ is defined.

2. Find the domain of the outer function $f(x)$:

   • This involves ensuring that the values output by $g(x)$ (i.e., the inputs to $f$) must be valid for $f(x)$. So, find where $f$ is defined.

3. Combine the conditions:

   • The domain of $f(g(x))$ is all $x$-values for which $g(x)$ is defined and for which $f(g(x))$ is defined.

Example:

Let $f(x) = \sqrt{x}$ and $g(x) = x^2 - 1$. Find the domain of $(f \circ g)(x) = f(g(x)) = \sqrt{x^2 - 1}$.

1. Domain of the inner function $g(x) = x^2 - 1$:

   • $g(x)$ is a polynomial, so it is defined for all real numbers, i.e., $(-\infty, \infty)$.

2. Domain of the outer function $f(x) = \sqrt{x}$:

   • The square root function $f(x) = \sqrt{x}$ is defined only when $x \geq 0$.

3. Combine the conditions:

   • We need $g(x) = x^2 - 1 \geq 0$, because $f(g(x)) = \sqrt{x^2 - 1}$.
   • Solve $x^2 - 1 \geq 0$: $x^2 \geq 1 \quad \Rightarrow \quad x \geq 1 \text{ or } x \leq -1$
   • So, the domain of $(f \circ g)(x)$ is $(-\infty, -1] \cup [1, \infty)$.

Conclusion:

The domain of the composition $(f \circ g)(x)$ is $(-\infty, -1] \cup [1, \infty)$.

Would you like further details or have any questions? Here are some related questions to expand this understanding:

1. What happens if the outer function has a more restrictive domain?
2. How do you handle compositions with rational functions?
3. Can you find the domain for $(f \circ g)(x)$ when $g(x)$ is a trigonometric function?
4. What is the domain of a composition where the outer function is a logarithmic function?
5. How does the range of the inner function affect the composition's domain?

Tip: Always pay close attention to whether the inner function produces values that are compatible with the domain of the outer function.