The problem asks you to state the domain and range of two transformed functions of $f(x)$:

i) $y = -2f(x)$
ii) $y = \frac{1}{2}f(x)$

General Considerations:

• Domain: The domain of a function remains the same regardless of vertical transformations (such as multiplying by a constant or flipping the graph vertically), unless there is a restriction caused by the function itself.
• Range: Vertical transformations affect the range of a function. A multiplication of the function by a positive constant stretches the range, while multiplying by a negative constant reflects the range vertically.

Let’s analyze each function separately.

i) $y = -2f(x)$

• The domain of $y = -2f(x)$ is the same as that of $f(x)$, because multiplying by $-2$ doesn’t change the input values.
• The range is affected. Multiplying $f(x)$ by $-2$ reflects it across the x-axis (because of the negative sign) and stretches the values vertically by a factor of 2. If the original range of $f(x)$ was $[a, b]$, then the new range would be $[-2b, -2a]$, assuming $a < b$.

ii) $y = \frac{1}{2}f(x)$

• The domain of $y = \frac{1}{2}f(x)$ is also the same as that of $f(x)$, since multiplying by $\frac{1}{2}$ does not affect the input values.
• The range is scaled vertically by a factor of $\frac{1}{2}$. If the original range of $f(x)$ was $[a, b]$, the new range would be $\left[ \frac{a}{2}, \frac{b}{2} \right]$.

To provide exact domain and range values, we would need the original function $f(x)$ and its properties, such as its domain and range.

Would you like to go into more detail about transformations of functions? Let me know if you have specific questions!

Related Questions:

1. What happens to the range if you add or subtract a constant from $f(x)$?
2. How does the graph of $f(x)$ change if $x$ is scaled by a constant instead of $f(x)$?
3. What is the effect on the domain if you shift $f(x)$ horizontally?
4. How can you find the domain and range of a composition of functions like $f(g(x))$?
5. What if $f(x)$ is a piecewise function—how does that affect the transformations?

Tip:

To fully understand transformations, it helps to sketch the graph of $f(x)$ and apply changes one step at a time to see the effect on both the domain and range.