Define the function formula for each of the following given the function definitions for 
f
, 
g
, and 
h
.

f
(
x
)
=
√
x
+
5
g
(
x
)
=
3
x
−
6

h
(
x
)
=
5
x
f
(
g
(
x
)
)
=
sqrt(3x-6+5)
 Correct   

g
(
f
(
x
)
)
=
3√x + 9
 Incorrect    
3
√
x
+
9
 syntax ok. warning: [use function notation - use sqrt(x) instead of sqrtx].

h
(
f
(
x
)
)

Let's go through each function and the compositions of those functions based on the provided definitions:

1. **Given functions:**
   - \( f(x) = \sqrt{x} + 5 \)
   - \( g(x) = 3x - 6 \)
   - \( h(x) = 5x \)

2. **Composition 1:**
   \( f(g(x)) \)

   This composition means you substitute \( g(x) \) into \( f(x) \).
   
   - Since \( f(x) = \sqrt{x} + 5 \), and \( g(x) = 3x - 6 \), substitute \( g(x) \) into \( f(x) \):
     \[
     f(g(x)) = \sqrt{g(x)} + 5 = \sqrt{3x - 6} + 5
     \]
   This is correct.

3. **Composition 2:**
   \( g(f(x)) \)

   This means you substitute \( f(x) \) into \( g(x) \).
   
   - Since \( g(x) = 3x - 6 \), and \( f(x) = \sqrt{x} + 5 \), substitute \( f(x) \) into \( g(x) \):
     \[
     g(f(x)) = 3f(x) - 6 = 3(\sqrt{x} + 5) - 6 = 3\sqrt{x} + 15 - 6 = 3\sqrt{x} + 9
     \]
   This is the correct expression.

4. **Composition 3:**
   \( h(f(x)) \)

   This means you substitute \( f(x) \) into \( h(x) \).
   
   - Since \( h(x) = 5x \), and \( f(x) = \sqrt{x} + 5 \), substitute \( f(x) \) into \( h(x) \):
     \[
     h(f(x)) = 5f(x) = 5(\sqrt{x} + 5) = 5\sqrt{x} + 25
     \]
   So, \( h(f(x)) = 5\sqrt{x} + 25 \).

---

### Summary of the function compositions:
1. \( f(g(x)) = \sqrt{3x - 6} + 5 \)
2. \( g(f(x)) = 3\sqrt{x} + 9 \)
3. \( h(f(x)) = 5\sqrt{x} + 25 \)

Would you like more details or have any questions about the process?

Here are 5 questions that expand on the topic:

1. What is the domain of \( f(g(x)) \)?
2. How would you find \( g(h(x)) \)?
3. Can you compute \( f(f(x)) \) using the given function?
4. What would be the inverse of \( f(x) \)?
5. How do you graph \( h(f(x)) \)?

**Tip:** When dealing with function compositions, always carefully substitute the entire inner function into the outer function for clarity and accuracy.

Define the function formula for each of the following given the function definitions for f(x) = √x + 5, g(x) = 3x − 6, and h(x) = 5x. Find the compositions f(g(x)), g(f(x)), and h(f(x)).

Explore how to solve function compositions using the functions f(x) = √x + 5, g(x) = 3x − 6, and h(x) = 5x. Learn how to find f(g(x)), g(f(x)), and h(f(x)) with a step-by-step breakdown. Suitable for high school students in Grades 9-12.

Math Problem Statement

f ( x )

√ x + 5 g ( x )

h ( x )

5 x f ( g ( x ) )

g ( f ( x ) )

Solution

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