Find (*f*∘*g*)(*x*) and (*g*∘*f*)(*x*)

 for the pair of functions.
 
*f*(*x*)=*x*+13/2

, *g*(*x*)=√4−*x*

 

We are asked to find the composite functions \((f \circ g)(x)\) and \((g \circ f)(x)\) for the given pair of functions:

- \( f(x) = x + \frac{13}{2} \)
- \( g(x) = \sqrt{4 - x} \)

### 1. Finding \( (f \circ g)(x) \):

This represents \( f(g(x)) \), which means we substitute \( g(x) \) into \( f(x) \).

- First, recall \( g(x) = \sqrt{4 - x} \).
- Now substitute \( g(x) \) into \( f(x) \):

\[
f(g(x)) = f(\sqrt{4 - x}) = \sqrt{4 - x} + \frac{13}{2}
\]

Thus,

\[
(f \circ g)(x) = \sqrt{4 - x} + \frac{13}{2}
\]

### 2. Finding \( (g \circ f)(x) \):

This represents \( g(f(x)) \), which means we substitute \( f(x) \) into \( g(x) \).

- First, recall \( f(x) = x + \frac{13}{2} \).
- Now substitute \( f(x) \) into \( g(x) \):

\[
g(f(x)) = g\left(x + \frac{13}{2}\right) = \sqrt{4 - \left(x + \frac{13}{2}\right)}
\]

Simplify the expression inside the square root:

\[
g(f(x)) = \sqrt{4 - x - \frac{13}{2}} = \sqrt{\frac{8}{2} - x - \frac{13}{2}} = \sqrt{\frac{-5}{2} - x}
\]

Thus,

\[
(g \circ f)(x) = \sqrt{\frac{-5}{2} - x}
\]

### Summary:
- \( (f \circ g)(x) = \sqrt{4 - x} + \frac{13}{2} \)
- \( (g \circ f)(x) = \sqrt{\frac{-5}{2} - x} \)

Do you need further details or have any questions?

Here are 5 related questions to explore:
1. What is the domain of \( f(x) \) and \( g(x) \)?
2. How do you find the domain of composite functions?
3. What happens if the result inside the square root is negative?
4. Can you combine the results of \( f(x) \) and \( g(x) \) in a different way?
5. How does composition change if \( g(x) \) is more complex?

**Tip:** Always check the domains when working with composite functions to avoid invalid inputs like negative square roots.

Find (f∘g)(x) and (g∘f)(x) for the pair of functions f(x) = x + 13/2 and g(x) = √(4−x)

Learn how to solve for the composite functions (f∘g)(x) and (g∘f)(x) using the functions f(x) = x + 13/2 and g(x) = √(4 - x). This solution covers key algebraic techniques involving radical expressions and function composition. Suitable for high school students in Grades 10-12.

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