Where Graphs Meet


Suppose we sketch the graphs of two functions $f(x)$ and $g(x)$ on the same axes. The $x$-coordinates of points where the graphs meet are the solutions to the equation $f(x)=g(x)$.
We can use this property to solve equations graphically, but we must make sure the graphs are drawn carefully and accurately.

Let’s take a look at the following graphs of $y=-x+k$ and $y=\dfrac{1}{x}$.

Two graphs have two intersections when $k=3$.

Two graphs have one intersection when $k=2$.

Two graphs do not intersect when $k=1$.

Two graphs do not intersect when $k=0$.

Two graphs do not intersect when $k=-1$.

Two graphs have one intersection when $k=-2$.

Two graphs have two intersections when $k=-3$.

We can now easily summarise the following from above cases.
\begin{array}{|c|c|} \hline
k \lt -2 \text{ or } k \gt 2 & \text{two intersections} \\ \hline
k = -2 \text{ or } k = 2 & \text{one intersection} \\ \hline
-2 \lt k \lt 2 & \text{no intersection} \\ \hline
\end{array}


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