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To determine if \( f(x) \) is the inverse of \( g(x) \), we must check if the composition of the two functions, \( f(g(x)) \) and \( g(f(x)) \), results in \( x \). In other words, if both \( f(g(x)) = x \) and \( g(f(x)) = x \), then \( f(x) \) and \( g(x) \) are inverses of each other.

### Given:
\[
f(x) = -\left(\frac{1}{4} \cdot \sqrt[3]{x + 9}\right) - \frac{7}{4}
\]
\[
g(x) = (-4x - 7)^3 - 9
\]

### Step 1: Find \( f(g(x)) \)

Substitute \( g(x) = (-4x - 7)^3 - 9 \) into \( f(x) \):

\[
f(g(x)) = -\left(\frac{1}{4} \cdot \sqrt[3]{(-4x - 7)^3 - 9 + 9}\right) - \frac{7}{4}
\]

Simplify:
\[
f(g(x)) = -\left(\frac{1}{4} \cdot \sqrt[3]{(-4x - 7)^3}\right) - \frac{7}{4}
\]
\[
f(g(x)) = -\left(\frac{1}{4} \cdot (-4x - 7)\right) - \frac{7}{4}
\]
\[
f(g(x)) = -\left(\frac{-4x - 7}{4}\right) - \frac{7}{4}
\]
\[
f(g(x)) = x
\]

### Step 2: Find \( g(f(x)) \)

Substitute \( f(x) = -\left(\frac{1}{4} \cdot \sqrt[3]{x + 9}\right) - \frac{7}{4} \) into \( g(x) \):

\[
g(f(x)) = (-4 \left( -\left(\frac{1}{4} \cdot \sqrt[3]{x + 9}\right) - \frac{7}{4} \right) - 7)^3 - 9
\]

Simplify:
\[
g(f(x)) = \left( 4 \cdot \frac{1}{4} \cdot \sqrt[3]{x + 9} + 7 - 7 \right)^3 - 9
\]
\[
g(f(x)) = \left( \sqrt[3]{x + 9} \right)^3 - 9
\]
\[
g(f(x)) = x + 9 - 9
\]
\[
g(f(x)) = x
\]

### Conclusion:

Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), \( f(x) \) is the inverse of \( g(x) \). The answer is **"Yes"**.

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. How can you algebraically derive the inverse of a function?
2. What happens when two functions are not inverses of each other?
3. How can you verify the inverse graphically?
4. What is the role of composition of functions in verifying inverses?
5. How does the domain and range affect the inverse of a function?

**Tip:** When dealing with cubic functions and cube roots, simplifying step-by-step is essential to avoid mistakes.

Is f(x) the inverse function of g(x)? Given f(x) = -(1/4) * (cube root of (x + 9)) - 7/4 and g(x) = (-4x - 7)^3 - 9.

Learn how to verify if two functions are inverses by composing f(x) and g(x). This problem covers algebraic techniques involving cubic functions and cube roots, and checks if f(x) and g(x) satisfy the conditions for inverse functions. Suitable for high school students in Grades 10-12.

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