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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: \( [0, \infty) \)

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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.

Given functions f(t) = sqrt(t) and g(t) = 1 - t^5, construct the composite functions (f ∘ g)(t) and (g ∘ f)(t). Then, determine each composite function's domain using interval notation.

Learn how to construct composite functions and determine their domains using interval notation. This problem involves the functions f(t) = sqrt(t) and g(t) = 1 - t^5, covering key concepts in composite and radical functions. Suitable for students in Grades 11-12.

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