Let’s analyze the given function $f(x) = -\sqrt[3]{x - 7} - 4$ and solve the problem.

Part (a) Sketch a graph of the function

The function $f(x) = -\sqrt[3]{x - 7} - 4$ is a transformation of the parent cube root function $y = \sqrt[3]{x}$. Here's how the transformations work:

1. Horizontal shift by +7: The term $x - 7$ shifts the graph 7 units to the right.
2. Vertical reflection: The negative sign in front of $\sqrt[3]{x - 7}$ reflects the graph across the x-axis.
3. Vertical shift by -4: The term $-4$ shifts the graph 4 units downward.

The general behavior of the cube root function remains: it is symmetric around its inflection point (here at $(7, -4)$), with the graph decreasing to the left and right of this point.

Part (b) Solve for $x$ when $f(x) = -2$

To solve $f(x) = -2$, substitute $-2$ into the function:

$-2 = -\sqrt[3]{x - 7} - 4$

1. Isolate the cube root term: $-2 + 4 = -\sqrt[3]{x - 7}$

$2 = -\sqrt[3]{x - 7}$

$-2 = \sqrt[3]{x - 7}$

2. Cube both sides to eliminate the cube root: $(-2)^3 = x - 7$

$-8 = x - 7$

3. Solve for $x$: $x = -8 + 7$

$x = -1$

Final Answer:

• (a) The graph is a cube root curve shifted and reflected as described.
• (b) $x = -1$ when $f(x) = -2$.

Would you like me to provide:

1. A sketch of the graph?
2. A more detailed explanation of the transformations?
3. Graphing tips or further examples?
   Let me know!

Questions to Explore Further:

1. How do horizontal and vertical shifts affect other parent functions (e.g., quadratic or sine)?
2. What are the differences between cube root and square root functions in terms of their domain and range?
3. How does reflection across the x-axis affect symmetry in other functions?
4. What happens to the graph if the cube root is multiplied by a constant $a > 1$?
5. How would the solution to $f(x) = c$ differ if $c > -4$?

Tip:

For cube root transformations, always locate the inflection point first (here at $(7, -4)$) and then apply transformations step by step.