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We are given two functions, \( f(x) = \sqrt{x} \) and \( g(x) = 4x + 5 \), and we need to find and analyze the following composite functions:

1. \( f \circ g \) (or \( f(g(x)) \))
2. \( g \circ f \) (or \( g(f(x)) \))
3. \( f \circ f \) (or \( f(f(x)) \))
4. \( g \circ g \) (or \( g(g(x)) \))

Let's compute each composite function and its domain:

### (a) \( f \circ g \) (or \( f(g(x)) \))
Here, \( f(g(x)) = f(4x + 5) = \sqrt{4x + 5} \).

**Domain**: The expression \( 4x + 5 \geq 0 \) must hold for the square root to be defined.
\[
4x + 5 \geq 0 \implies x \geq -\frac{5}{4}.
\]
Thus, the domain is \( x \in \left[ -\frac{5}{4}, \infty \right) \).

### (b) \( g \circ f \) (or \( g(f(x)) \))
Here, \( g(f(x)) = g(\sqrt{x}) = 4\sqrt{x} + 5 \).

**Domain**: Since \( \sqrt{x} \) is only defined for \( x \geq 0 \), the domain of \( g(f(x)) \) is \( x \in [0, \infty) \).

### (c) \( f \circ f \) (or \( f(f(x)) \))
Here, \( f(f(x)) = f(\sqrt{x}) = \sqrt{\sqrt{x}} = x^{1/4} \).

**Domain**: The square root is defined for \( x \geq 0 \), so the domain is \( x \in [0, \infty) \).

### (d) \( g \circ g \) (or \( g(g(x)) \))
Here, \( g(g(x)) = g(4x + 5) = 4(4x + 5) + 5 = 16x + 20 + 5 = 16x + 25 \).

**Domain**: The function \( g(g(x)) \) is a linear function, which is defined for all real values of \( x \). Thus, the domain is \( x \in (-\infty, \infty) \).

### Summary of Domains:
- \( f \circ g \): \( x \in \left[ -\frac{5}{4}, \infty \right) \)
- \( g \circ f \): \( x \in [0, \infty) \)
- \( f \circ f \): \( x \in [0, \infty) \)
- \( g \circ g \): \( x \in (-\infty, \infty) \)

Would you like more details or have any questions on these composite functions?

Here are some related questions to explore:
1. How do you find the domain of a composite function in general?
2. What happens when you compose two functions that have restricted domains?
3. How do transformations of the square root function affect its graph?
4. Can you always compose two functions regardless of their domains?
5. How do linear functions behave under composition?

**Tip**: When working with square root functions, always check for non-negative values inside the root to determine the domain properly.

For f(x) = √x and g(x) = 4x + 5, find the following composite functions and state the domain of each: (a) f ∘ g, (b) g ∘ f, (c) f ∘ f, (d) g ∘ g.

Learn how to compute the composite functions f ∘ g, g ∘ f, f ∘ f, and g ∘ g for f(x) = √x and g(x) = 4x + 5. Explore the domains of these functions, considering restrictions for square root and linear expressions. Suitable for high school students.

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