Let's analyze the graph step by step for each question: --- ## (a) **The domain of f** - The domain is the set of all x-values for which the function is defined. - From the graph, the curve starts at \( x = -8 \) and ends at \( x = 2 \) (closed dot at \( x = 2 \)), so: - **Domain:** \(\boxed{[-8, 2]}\) --- ## (b) **The range of f** - The range is the set of all y-values that the function takes. - The lowest point on the graph is at \( y = -6 \) and the highest at \( y = 6 \): - **Range:** \(\boxed{[-6, 6]}\) --- ## (c) **The zeros of f** - Zeros are the x-values where \( f(x) = 0 \) (where the graph intersects the x-axis). - From the graph, the function crosses the x-axis at \( x = -6 \) and \( x = 1 \): - **Zeros:** \(\boxed{x = -6, 1}\) --- ## (d) **f(0)** - The value of the function when \( x = 0 \). - From the graph, at \( x = 0 \), \( y = 2 \): - \(\boxed{f(0) = 2}\) --- ## (e) **The intervals on which f is increasing** - The function is increasing where the graph goes up as you move from left to right. - From \( x = -8 \) to \( x = -2 \), the graph increases: - **Increasing on:** \(\boxed{[-8, -2]}\) --- ## (f) **The intervals on which f is decreasing** - The function is decreasing where the graph goes down as you move from left to right. - From \( x = -2 \) to \( x = 2 \), the graph decreases: - **Decreasing on:** \(\boxed{[-2, 2]}\) --- ## (g) **The values for which \( f(x) \leq 0 \)** - These are the x-values where the graph is on or below the x-axis. - From the graph, \( f(x) \leq 0 \) for \( x \in [-8, -6] \cup [1, 2] \): - \(\boxed{[-8, -6] \cup [1, 2]}\) --- ## (h) **Any relative maxima or minima** - **Relative maximum:** The highest point in a local region. - The maximum occurs at \( x = -2 \), where \( y = 6 \). - **Relative minima:** The lowest point in a local region. - There are no relative minima in the interior (the endpoints are not considered relative minima since the function doesn't "turn" there). - **Relative maximum:** \(\boxed{( -2, 6 )}\) - **Relative minimum:** \(\boxed{\text{None}}\) --- ## (i) **The value(s) of x for which \( f(x) = 4 \)** - Where does the graph cross \( y = 4 \)? - From the graph, \( f(x) = 4 \) at \( x = -4 \) and \( x = 0 \): - \(\boxed{x = -4, 0}\) --- ## (j) **Is \( f(-2) \) positive or negative?** - From the graph, at \( x = -2 \), \( y = 6 \) (which is positive). - **Positive:** \(\boxed{\text{Positive}}\) --- ## **Summary Table** | Part | Answer | |------|---------------------------------------| | a | \([-8, 2]\) | | b | \([-6, 6]\) | | c | \(x = -6, 1\) | | d | 2 | | e | \([-8, -2]\) | | f | \([-2, 2]\) | | g | \([-8, -6] \cup [1, 2]\) | | h | Max: \((-2, 6)\); Min: None | | i | \(x = -4, 0\) | | j | Positive |