Question - Graph Analysis of a Function

\textbf{(a) The domain of } f:

[\text{All real numbers}] \text{, since the graph extends infinitely in the x-direction.}

\textbf{(b) The range of } f:

[-3, \infty) \text{, because the highest y-value the graph reaches is infinite and the lowest is } -3.

\textbf{(c) The zeros of } f:

\{ -4, 2 \} \text{, the x-values where the graph intersects the x-axis.}

\textbf{(d) } f(-3.5):

\text{As } x = -3.5, \text{ f(x) is about } 2.5 \text{, reading from the graph.}

\textbf{(e) The intervals on which } f \text{ is increasing:}

(-\infty, -4) \cup (2, \infty) \text{, the intervals on the x-axis where the graph goes upwards as x increases.}

\textbf{(f) The intervals on which } f \text{ is decreasing:}

(-4, 2) \text{, the interval on the x-axis where the graph goes downwards as x increases.}

\textbf{(g) The values for which } f(x) \leq 0:

[-4, 2] \text{, these are the x-values where the graph is at or below the x-axis.}

\textbf{(h) Any relative maxima or minima:}

\text{Relative maximum at } (2, 3) \text{, relative minimum at } (-4, -3) \text{ based on the graph's high and low points respectively.}

\textbf{(i) The value(s) of } x \text{ for which } f(x) = -3:

\{-4, 0\} \text{, the x-values where the graph touches the horizontal line } y = -3.

\textbf{(j) Is } f(0) \text{ positive or negative?}

\text{Negative, since the point } (0, f(0)) \text{ lies below the x-axis where } f(0) \text{ is around } -3.