The question provided asks to construct the composite functions $(f \circ g)(t)$ and $(g \circ f)(t)$ for the given functions:

$f(t) = \sqrt{t}$ $g(t) = 1 - t^5$

After constructing the composites, we are also asked to determine the domain of each composite function using interval notation.

1. Composite Function $(f \circ g)(t)$

This represents $f(g(t))$. To construct it, substitute $g(t)$ into $f(t)$:

$(f \circ g)(t) = f(g(t)) = f(1 - t^5) = \sqrt{1 - t^5}$

For the domain, the expression inside the square root, $1 - t^5$, must be non-negative:

$1 - t^5 \geq 0$ $t^5 \leq 1$ $t \leq 1$

Thus, the domain for $(f \circ g)(t)$ is $(-\infty, 1]$.

2. Composite Function $(g \circ f)(t)$

This represents $g(f(t))$. To construct it, substitute $f(t)$ into $g(t)$:

$(g \circ f)(t) = g(f(t)) = g(\sqrt{t}) = 1 - (\sqrt{t})^5 = 1 - t^{5/2}$

For the domain, we require that $t \geq 0$ because $f(t) = \sqrt{t}$ requires non-negative values of $t$.

Thus, the domain for $(g \circ f)(t)$ is $[0, \infty)$.

Final Answer:

1. $(f \circ g)(t) = \sqrt{1 - t^5}$, domain: $(-\infty, 1]$
2. $(g \circ f)(t) = 1 - t^{5/2}$, domain: $[0, \infty)$

Would you like further details or have any questions?

Here are 5 relative questions to expand on this information:

1. How would the domain change if $g(t)$ involved a higher degree polynomial?
2. What are the steps for finding the range of composite functions?
3. How would the composition change if $f(t)$ were a rational function?
4. Can the domain of a composite function be larger than that of its components?
5. How would you find the inverse of the composite function $(f \circ g)(t)$?

Tip: Always consider both the inner and outer functions when determining the domain of a composite function, as both influence the valid input values.