Let’s analyze and solve each part of the given problem step by step:

Question Overview

The curve $y = f(x)$ is a cubic polynomial. The diagram shows the curve, and the following tasks are asked:

1. (a) Find the values of $x$ for which $f(x) < \frac{1}{2}$, giving the answer in set notation.
2. (b) Sketch the graph of $y = f(-x)$.
3. (c) Explain why $f(x)$ cannot be expressed as the product of three real linear factors.

Part (a): $f(x) < \frac{1}{2}$

1. Locate where $f(x) = \frac{1}{2}$ on the graph:

   • The horizontal line $y = \frac{1}{2}$ intersects the curve at two points:
     • $x \approx -1.7$ (left of the origin).
     • $x \approx 2.3$ (right of the origin).

2. Analyze the regions where $f(x) < \frac{1}{2}$:

   • Between $x \approx -1.7$ and $x = 1$, the curve lies below $y = \frac{1}{2}$.

3. Answer in set notation: $x \in (-1.7, 1)$

Part (b): Sketching $y = f(-x)$

The graph of $y = f(-x)$ reflects $y = f(x)$ across the $y$-axis. Key points to consider:

1. Reflect the $x$-coordinates of key points:

   • The local maximum at $(2, 1)$ becomes $(-2, 1)$.
   • The local minimum at $(1, -1)$ becomes $(-1, -1)$.
   • The point $(3, 2)$ becomes $(-3, 2)$.

2. Sketch the symmetric curve, ensuring that the overall shape of the cubic function remains consistent but mirrored.

Part (c): Why $f(x)$ cannot be expressed as the product of three real linear factors

A cubic polynomial $f(x)$ can be expressed as the product of three real linear factors only if it has three distinct real roots.

1. The graph of $y = f(x)$ crosses the $x$-axis only once (at approximately $x \approx -3$).
2. Since $f(x)$ has only one real root, the remaining two roots must be complex.

Explanation: $f(x) \text{ cannot be expressed as the product of three real linear factors because it has only one real root.}$

Summary of Solutions

• (a) $x \in (-1.7, 1)$
• (b) Reflect the graph of $y = f(x)$ across the $y$-axis.
• (c) $f(x)$ has only one real root, so it cannot be expressed as the product of three real linear factors.

Would you like further clarification on any part or assistance in sketching the graph?

Further Questions

1. What is the general process for finding where $f(x) < k$ for other values of $k$?
2. How do we formally derive the points of intersection $f(x) = \frac{1}{2}$ using algebraic techniques?
3. What other transformations of the graph $y = f(x)$ can be applied (e.g., $y = -f(x), y = f(x+2)$)?
4. How do the roots of a polynomial relate to its factored form?
5. Why do cubic polynomials with one real root necessarily have complex conjugate roots?

Tip: When reflecting a function $f(x)$ to get $f(-x)$, flip the sign of each $x$-coordinate while keeping the $y$-coordinate unchanged.