Parabolas and Inverse Functions
Task
 Explain why the equation $y = x^2$ represents $y$ as a function of a real variable $x$.
 For the relation considered in part (a), is $x$ a function of $y$? Explain.

Give a context in which the equation $y = x^2$ does represent $x$ as a function of $y$.
IM Commentary
This problem is a simple decontextualized version of FIF Your Father and FIF Parking Lot. It also provides a natural context where the absolute value function arises as, in part (b), solving for $x$ in terms of $y$ means taking the square root of $x^2$ which is $x$.
This task assumes students have an understanding of the relationship between functions and equations. Using this knowledge, the students are prompted to try to solve equations in order to find the inverse of a function given in equation form: when no such solution is possible, this means that the function does not have an inverse. Part (c) is an openended question which, with teacher guidance, leads students to realize that this problem can be fixed by restricting the domain of the function to a smaller set.
Solution
 The equation $y = x^2$ makes $y$ a function of $x$ because for each real number $x_0$, the equation $y = x^2$ assigns $y$ a value of $x_0^2$. So the equation $y = x^2$ naturally gives rise to the function $f$ defined by the rule $f(x) = x^2$.

For most values of $y$, such as $y = 1$, there are two values of $x$, $x = 1$ and $x = 1$ in this case, which satisfy $y = x^2$. If we could write $x$ as a function of $y$ then each $y$ value would correspond to only one $x$ value. So $x$ cannot be written as a function of $y$ if we try to do this for all real numbers $x$. The relation in part (a) is for all real numbers $x$ and so $x$ is not a function of $y$ in this setting.

Looking at part (b), $x$ can be viewed as a function of $y$ as long as we do not include pairs of values of equal distance from $0$, such as $x = +1$ and $x = 1$, in the set of $x$ values. For example, if $x$ is restricted to the nonnegative real numbers, we have the equation $x = \sqrt{y}$ describing $x$ as a function of $y$. Another way to express this is the following: when $x$ is restricted to the nonnegative real numbers, each nonnegative $y$ value corresponds to exactly one nonnegative $x$ value so $x$ can be viewed as a function of $y$. Similarly, if $x$ is restricted to the nonpositive real numbers, we can represent $x$ as a function of $y$ by the equation $x=\sqrt{y}$.
In fact, the set of $x$values considered could be quite complex  it need only avoid containing both the values $a$ and $a$ for a nonzero real number $a$. For example, the set of $x$values could be all real numbers greater than $3$ together with those (negative) real numbers between $2$ and $1$. The set could also be extremely simple: For example, if only the value $5$ is allowed for $x$ then the equation $y = x^2$ makes $x$ a function of $y$.
Parabolas and Inverse Functions
 Explain why the equation $y = x^2$ represents $y$ as a function of a real variable $x$.
 For the relation considered in part (a), is $x$ a function of $y$? Explain.

Give a context in which the equation $y = x^2$ does represent $x$ as a function of $y$.