1. **Problem Statement:** We have two ellipses on a Cartesian plane: one gray ellipse representing a function $f(x)$ and one black ellipse representing a function $g(x)$. The black ellipse is centered to the right of the y-axis, and the gray ellipse is centered to the left. We need to express $g(x)$ as a function of $f(x)$. 2. **Understanding Ellipses and Translations:** The general form of an ellipse centered at $(h,k)$ is: $$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$ If the gray ellipse is centered at $(-c,0)$ and the black ellipse at $(c,0)$ for some $c>0$, then the black ellipse is a horizontal translation of the gray ellipse by $2c$ units to the right. 3. **Expressing $g(x)$ in terms of $f(x)$:** If $f(x)$ is the function describing the gray ellipse, then shifting it horizontally by $2c$ units to the right gives: $$g(x) = f(x - 2c)$$ This means for every $x$, the value of $g(x)$ equals the value of $f$ at $x-2c$. 4. **Summary:** The black ellipse function $g(x)$ is the horizontal translation of the gray ellipse function $f(x)$ by $2c$ units to the right: $$\boxed{g(x) = f(x - 2c)}$$ This relationship holds assuming the ellipses have the same shape and size, differing only by horizontal translation.