# A Sum of Functions

## Task

Using the graphs below, sketch a graph of the function $s(x) = f(x) + g(x)$.

## IM Commentary

The intent of this problem is to have students think about how function addition works on a fundamental level, so formulas have been omitted on purpose. In the graph shown, $g(x)=\frac{4}{x^2+1}$. The task may be extended by asking students to sketch the graph of $d(x)=f(x)-g(x)$.

Although this problem does not ask students to "write a function that describes a relationship between two quantities", it can provide students with understandings preparatory for F.BF.1b. In addition, this task makes use of the reasoning required for F.BF.3.

Source: Hilton Russell

## Solutions

Solution: Graphical solution

Students can create the graph shown below by:

- visually estimating the distance between the graph of $f$ and the $x$-axis at a particular integer value of $x$, and
- plotting a point this distance above (or below, if the $f(x)$ value is negative) the graph of $g$.

Some students may want to use a strip of paper to mark a distance and then use the mark to help them plot the point.

Solution: Numerical solution

Students may also create a chart of approximate values of $f(x)$ and $g(x)$ at various $x$-values by estimating from the provided graphs. We then add a row of $s(x)$ values by summing the two rows above. Finally, we plot points of the form $(x, s(x))$ to sketch the graph of $y=s(x)$.

$x$ | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |

$f(x)\approx$ | -1 | -0.5 | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |

$g(x)\approx$ | 0.2 | 0.4 | 0.8 | 2 | 4 | 2 | 0.8 | 0.4 | 0.2 |

$s(x)=f(x) + g(x)\approx$ | -0.8 | -0.1 | 0.8 | 2.5 | 5 | 3.5 | 2.8 | 2.9 | 3.2 |

## A Sum of Functions

Using the graphs below, sketch a graph of the function $s(x) = f(x) + g(x)$.