Points on a graph

Task

Suppose $f$ is a function.

1. If $10=f(-4)$, give the coordinates of a point on the graph of $f$.
2. If $6$ is a solution of the equation $f(w)=1$, give a point on the graph of $f$.

IM Commentary

This task is designed to get at a common student confusion between the independent and dependent variables. This confusion often arises in situations like (b), where students are asked to solve an equation involving a function, and confuse that operation with evaluating the function.

This task is adapted from Functions Modeling Chanage: A Preparation for Calculus, Connally et al., Wiley 2010.

Solution

1. The graph of a function is the set of ordered pairs $(x,f(x))$, with $x$ ranging over the domain of $f$. So the point $(-4,f(-4))$ is on the graph of $f$, and since $10 = f(-4)$, this means that the point $(-4,10)$ is on the graph.
2. If $6$ is a solution to $f(w) = 1$, that means that $f(6) = 1$, so the point $(6,1)$ is on the graph of $f$.