if f(x)=sqrt:x and g(x)= -3x-2, find (fog)(x) and (gof)(x)

Given the functions:

- \( f(x) = \sqrt{x} \)
- \( g(x) = -3x - 2 \)

We need to find both:

1. \( (f \circ g)(x) = f(g(x)) \)
2. \( (g \circ f)(x) = g(f(x)) \)

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

Substitute \( g(x) \) into \( f(x) \):

\[
(f \circ g)(x) = f(g(x)) = f(-3x - 2)
\]
Now, apply \( f(x) = \sqrt{x} \) to the expression:

\[
(f \circ g)(x) = \sqrt{-3x - 2}
\]

So, 
\[
(f \circ g)(x) = \sqrt{-3x - 2}
\]

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

Substitute \( f(x) \) into \( g(x) \):

\[
(g \circ f)(x) = g(f(x)) = g(\sqrt{x})
\]
Now, apply \( g(x) = -3x - 2 \) to \( \sqrt{x} \):

\[
(g \circ f)(x) = -3\sqrt{x} - 2
\]

Thus,
\[
(g \circ f)(x) = -3\sqrt{x} - 2
\]

### Final Answer:
- \( (f \circ g)(x) = \sqrt{-3x - 2} \)
- \( (g \circ f)(x) = -3\sqrt{x} - 2 \)

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

Here are five related questions to deepen understanding:

1. How do you compute the domain of \( (f \circ g)(x) \)?
2. What is the domain of \( (g \circ f)(x) \)?
3. Can the composition \( (f \circ g)(x) \) ever be negative?
4. What is the range of \( (f \circ g)(x) \) for specific values of \( x \)?
5. How does the behavior of these compositions change as \( x \to \infty \)?

**Tip**: Always verify the domain of composite functions to ensure that the operations are valid at each stage.

if f(x)=sqrt(x) and g(x)= -3x-2, find (fog)(x) and (gof)(x)

Learn how to solve the function composition problem involving f(x) = sqrt(x) and g(x) = -3x - 2. The step-by-step solution includes calculating (f ∘ g)(x) and (g ∘ f)(x) with detailed explanations of square root and linear functions. Suitable for high school students in Grades 9-12.

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