1. Problem: Determine which of the following figures represent the curve of a function in which its range $\neq$ its domain. 2. Definitions and rules: The domain is the set of x-values of points on the curve, written as $D=\{x:\,\exists y\text{ such that }(x,y)\text{ lies on the curve}\}$. The range is the set of y-values of points on the curve, written as $R=\{y:\,\exists x\text{ such that }(x,y)\text{ lies on the curve}\}$. Important rule: to compare domain and range we list the intervals or sets of x and y from the description and check equality of sets. 3. Analyze (a): The line starts at the origin and goes upward with positive slope, described as a ray beginning at $(0,0)$ and extending to the right. Interpreting this as $x\ge 0$ with corresponding $y\ge 0$, we have $D_a=[0,\infty)$ and $R_a=[0,\infty)$. Because $D_a=R_a$ this figure does not satisfy $R\neq D$. 4. Analyze (b): The line segment has open endpoints at $(-4,-10)$ and $(5,0)$ and passes through the origin, so the x-values run from -4 to 5 excluding endpoints. Thus $D_b=(-4,5)$ and the y-values run from -10 to 0 excluding endpoints, so $R_b=(-10,0)$. Since $(-4,5)\neq(-10,0)$ this figure satisfies $R\neq D$. 5. Analyze (c): The curve starts at the origin and increases steeply to the right, described as a ray from $(0,0)$ into the positive x and y directions. Restricting to the shown window a natural interpretation is $D_c=[0,4]$ and $R_c=[0,4]$ in that window. Under that interpretation $D_c=R_c$ and so (c) does not satisfy $R\neq D$. 6. Analyze (d): The curve starts near $(0,-10)$ at the bottom of the window and goes steeply downward to the right, so the x-values stay in a finite displayed interval while the y-values extend below -10 off the bottom of the window. A reasonable description is $D_d=[0,4]$ (displayed x-range) while $R_d$ extends below -10 (e.g. $(-\infty,-10)$), so $D_d\neq R_d$. Therefore (d) satisfies $R\neq D$. 7. Conclusion: The figures whose curves are functions with range not equal to domain are (b) and (d).