Where two functions meet

Solving f(x) = g(x) by graph, by table and by algebra, and checking the answer.

Two phone plans, two hire companies, two savings schemes: the useful question is almost never what does this cost, but at what point does the cheaper one stop being cheaper. That is a question about where two functions agree.

Put both functions in one table, with their difference beside them, and look for the moment the difference changes sign.

The difference runs 20, 16, 12, 8, 4, 0, -4, -8. It hits exactly zero at $x = 5$, and that is where the two functions agree. Before 5, $f$ is the larger. After 5, $g$ is.

That is the first of the three methods, and it also shows the idea underneath all of them: solving $f(x) = g(x)$ is the same as solving $f(x) - g(x) = 0$. The crossings of two graphs are the zeros of one function.

Method one: read it off the graphs

Draw both curves on the same axes and look at where they cross. Each crossing point gives one solution, and the solution is its horizontal coordinate. The vertical coordinate is the common value, which is often the second half of the question: at what number of units, and at what cost.

A graph is quick and it shows how many solutions there are, which the algebra can hide. It is also only as accurate as the drawing, so a graphical answer is usually a check or an estimate rather than the final word.

Method two: narrow it down numerically

When the crossing is not at a whole number, the table method can be repeated on a smaller interval. Watch it close in.

Each pass halves the interval, so after twelve passes the answer is pinned down to about a thousandth. This is how a calculator finds an intersection, and it is worth knowing that the number on the screen was found this way rather than exactly.

Method three: do the algebra

The exact answer comes from setting the two rules equal and rearranging. One worked example in full: solve $20 + 4x = 8x$.

1. Get the $x$ terms on one side: subtract $4x$ from both sides, giving $20 = 4x$.
2. Divide both sides by 4: $x = 5$.
3. Check by substituting into both original rules: $f(5) = 20 + 20 = 40$ and $g(5) = 40$.
4. They agree, so the crossing point is $(5, 40)$ and the answer is confirmed.

A second worked example, where the two functions are not both lines. Solve $x^2 = x + 6$.

1. Bring everything to one side: $x^2 - x - 6 = 0$.
2. The left-hand side is the difference function, so its zeros are the answers.
3. Testing whole numbers: at $x = 3$ it is $9 - 3 - 6 = 0$, and at $x = -2$ it is $4 + 2 - 6 = 0$.
4. Check both in the original equation: $3^2 = 9$ and $3 + 6 = 9$, then $(-2)^2 = 4$ and $-2 + 6 = 4$.
5. Both work, so the solutions are $x = 3$ and $x = -2$.