https://file.aiquickdraw.com/mathbot/file/17265218348482y5eu908.jpg

We are given the following functions:

- \( f(x) = \sqrt{x} \)
- \( g(x) = 4x - 1 \)
- \( h(x) = \frac{1}{x + 9} \)

We are tasked with finding \( (f \circ h \circ g)(x) \), which represents \( f(h(g(x))) \), and determining its domain.

### Step 1: Evaluate \( h(g(x)) \)

First, substitute \( g(x) = 4x - 1 \) into \( h(x) = \frac{1}{x + 9} \):
\[
h(g(x)) = h(4x - 1) = \frac{1}{(4x - 1) + 9} = \frac{1}{4x + 8}
\]

### Step 2: Evaluate \( f(h(g(x))) \)

Next, substitute \( h(g(x)) = \frac{1}{4x + 8} \) into \( f(x) = \sqrt{x} \):
\[
f(h(g(x))) = f\left( \frac{1}{4x + 8} \right) = \sqrt{\frac{1}{4x + 8}}
\]
So, \( f(h(g(x))) = \frac{1}{\sqrt{4x + 8}} \).

### Step 3: Determine the Domain

The domain is determined by the conditions that must be satisfied for the expression to be defined:

1. For the square root function \( \sqrt{4x + 8} \), we require \( 4x + 8 \geq 0 \), which simplifies to \( x \geq -2 \).
2. The denominator \( 4x + 8 \) cannot be zero because division by zero is undefined. Solving \( 4x + 8 = 0 \), we get \( x = -2 \), which must be excluded from the domain.

Thus, the domain is \( x > -2 \).

### Final Answer:
\[
(f \circ h \circ g)(x) = \frac{1}{\sqrt{4x + 8}}
\]
with the domain \( x > -2 \).

Would you like more details or have any questions?

---

Here are 5 related questions to further explore the topic:
1. How would the domain change if \( f(x) \) were defined as \( f(x) = x^2 \) instead of \( \sqrt{x} \)?
2. What is the range of \( (f \circ h \circ g)(x) \)?
3. How would you calculate the inverse of \( f(h(g(x))) \)?
4. How would you graph \( f(h(g(x))) \) and indicate its domain visually?
5. How does composition of functions affect the domain of the resulting function?

**Tip:** Always check both the inner and outer functions for domain restrictions when dealing with composite functions.

Let f(x) = sqrt(x), g(x) = 4x - 1, h(x) = 1/(x + 9). Find (f o h o g)(x) and its domain.

Solve the function composition (f o h o g)(x) with f(x) = sqrt(x), g(x) = 4x - 1, and h(x) = 1/(x + 9). This detailed solution covers composite functions, radicals, and domain restrictions. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation