Let's analyze the provided graph to answer the questions step by step.

(a) The domain of $f$

The domain consists of all possible input values (x-values) for which the function is defined. From the graph, we see that the curve starts around $x = -3$ and ends at $x = 6$, so the domain is:

$\text{Domain of } f = [-3, 6]$

(b) The range of $f$

The range consists of all possible output values (y-values) for the function. From the graph, the lowest point on the graph is around $y = -6$ and the highest point is near $y = 4$, so the range is:

$\text{Range of } f = [-6, 4]$

(c) The zeros of $f$

The zeros of a function occur where the graph crosses the x-axis (i.e., where $f(x) = 0$). From the graph, the function crosses the x-axis at approximately $x = 0$ and $x = 6$. Therefore, the zeros are:

$\text{Zeros of } f = \{0, 6\}$

(d) $f(5.5)$

To find $f(5.5)$, we check the graph at $x = 5.5$. At this point, the function appears to be slightly negative, around $y = -1$. Thus:

$f(5.5) \approx -1$

(e) The intervals on which $f$ is increasing

A function is increasing where its graph is rising from left to right. From the graph, the function is increasing from $x = -3$ to approximately $x = 2$. Thus, the interval where the function is increasing is:

$\text{Increasing on } (-3, 2)$

(f) The intervals on which $f$ is decreasing

A function is decreasing where its graph is falling from left to right. From the graph, the function is decreasing from $x = 2$ to $x = 6$. Thus, the interval where the function is decreasing is:

$\text{Decreasing on } (2, 6)$

(g) The values for which $f(x) \leq 0$

The function is less than or equal to 0 where the graph is either on or below the x-axis. From the graph, this occurs between $x = 0$ and $x = 6$. Thus:

$f(x) \leq 0 \text{ on } [0, 6]$

(h) Any relative maxima or minima

• Relative Maximum: The graph reaches a peak (relative maximum) at $x = 2$, where the y-value is approximately 4. So the relative maximum is: $\text{Relative Maximum: } (2, 4)$

• Relative Minimum: The graph has a relative minimum at $x = -3$, where the y-value is approximately $-6$. So the relative minimum is: $\text{Relative Minimum: } (-3, -6)$

(i) The values of $x$ for which $f(x) = 3$

From the graph, $f(x) = 3$ at two points: approximately $x = 1.5$ and $x = 4.5$. Therefore:

$f(x) = 3 \text{ at } x = 1.5 \text{ and } x = 4.5$

(j) Is $f(4)$ positive or negative?

At $x = 4$, the graph shows that the function is above the x-axis, meaning $f(4)$ is positive.

$f(4) \text{ is positive}$

Would you like further details or clarifications on any of these points?

5 Related Questions:

1. How can you algebraically confirm the zeros of a function from its equation?
2. What is the significance of the domain and range in real-life applications?
3. How can the intervals of increase or decrease be found using calculus?
4. How do you identify relative maxima and minima using the first or second derivative tests?
5. Can the behavior of the function outside the given domain or range be predicted?

Tip:

When analyzing graphs of functions, always check for symmetry and important features such as intercepts, maxima, and minima, as these can give you quick insights into the function's overall behavior.