1. **Problem Statement:** Determine if the function $f(x)$ described by the plot is one-to-one and onto. Also, find the domain and range of $f(x)$. 2. **One-to-One Function:** A function is one-to-one if each $y$-value corresponds to exactly one $x$-value. This means the function passes the horizontal line test: no horizontal line intersects the graph more than once. 3. **Onto Function:** A function is onto if every possible $y$-value in the codomain has at least one $x$-value in the domain such that $f(x) = y$. 4. **Domain:** The domain is the set of all $x$-values for which the function is defined. 5. **Range:** The range is the set of all $y$-values the function attains. 6. **Analysis of the Plot:** - The curve starts near $y=3$ at $x \approx -4$ and ends near $y=-3$ at $x \approx 3$. - The function decreases, crosses $y=0$ between $x=-2$ and $x=-1$. - It reaches a minimum below $y=-1$ between $x=-1$ and $0$. - Then it rises to a peak above $y=2$ between $x=1$ and $2$. - Finally, it falls steeply past $y=-3$ near $x=3$. 7. **Is $f$ one-to-one?** - Since the function has a minimum and a maximum, it fails the horizontal line test. - For example, a horizontal line between the minimum and maximum $y$-values will intersect the graph at multiple $x$-values. - Therefore, $f$ is **not one-to-one**. 8. **Is $f$ onto?** - The $y$-values covered by the function range from just below $-3$ to just above $3$. - Since the plot covers all $y$-values in this interval, the function is onto its codomain if the codomain is taken as this interval. - Therefore, $f$ is **onto** the interval approximately $[-3,3]$. 9. **Domain:** From the plot, $x$ ranges approximately from $-4$ to $3$. 10. **Range:** From the plot, $y$ ranges approximately from $-3$ to $3$. **Final answers:** - $f$ is **not one-to-one**. - $f$ is **onto** the interval $[-3,3]$. - Domain: $[-4,3]$. - Range: $[-3,3]$.