Let’s go through each part step-by-step using the information provided in the question.

Solution:

Part (a): Domain of $f$

The domain of $f$ includes all $x$-values for which the function $f(x)$ is defined. From the graph, the curve extends from $x = -\infty$ up to $x = 5.5$, where there’s a solid endpoint. Therefore, the domain is: $(-\infty, 5.5]$

Part (b): Range of $f$

The range of $f$ includes all $y$-values that $f(x)$ can attain. The curve goes up to $y = 4$ and down to negative infinity as $x$ decreases. Therefore, the range is: $(-\infty, 4]$

Part (c): Zeros of $f$

The zeros of $f$ are the $x$-values where $f(x) = 0$. Observing the graph, the function intersects the $x$-axis at $x = -1.5$ and at $x = 5.5$. Therefore, the zeros are: $x = -1.5 \text{ and } x = 5.5$

Part (d): $f(-1)$

From the graph, at $x = -1$, the corresponding $y$-value is $1$. Thus: $f(-1) = 1$

Part (e): Intervals on which $f$ is increasing

The function is increasing where the graph is moving upwards as $x$ increases. From the graph, $f$ increases from $x = -\infty$ up to $x = 2$. Therefore, the interval on which $f$ is increasing is: $(-\infty, 2)$

Part (f): Intervals on which $f$ is decreasing

The function is decreasing where the graph is moving downwards as $x$ increases. From the graph, $f$ decreases from $x = 2$ to $x = 5.5$. Therefore, the interval on which $f$ is decreasing is: $(2, 5.5)$

Part (g): Values for which $f(x) \leq 0$

We are looking for intervals where the graph is at or below the $x$-axis (where $f(x) \leq 0$). From the graph, this occurs on the intervals $x \in (-\infty, -1.5]$ and $x \in [5.5]$. So, the intervals are: $(-\infty, -1.5] \cup \{5.5\}$

Part (h): Relative Maxima and Minima

• The relative maximum occurs at the peak of the curve. From the graph, the maximum occurs at $x = 2$, with $f(2) = 4$.
• There is no relative minimum visible on this segment of the graph.

So, the relative maximum is: $\text{Maximum at } (2, 4)$

Part (i): Values of $x$ for which $f(x) = 4$

The graph shows that $f(x) = 4$ at $x = 2$. Therefore: $x = 2$

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

Observing the graph at $x = 4$, the $y$-value is positive (it’s between 2 and 4). Therefore: $f(4) \text{ is positive.}$

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Would you like further clarification on any part, or have additional questions?

Related Questions

1. How do we determine the domain of a function from a graph?
2. What is the difference between absolute and relative maxima?
3. How can you identify intervals of increase or decrease from a function’s equation?
4. What does it mean for a function to be “positive” or “negative” over an interval?
5. How do you find zeros of a function if a graph is not provided?

Tip

When reading graphs, use key points like intercepts, peaks, and endpoints to quickly determine characteristics such as domain, range, and behavior of the function.