### Video Transcript

The following linear graph
represents a function 𝑔 of 𝑥 after a reflection in the 𝑦-axis. Find the original function 𝑓 of
𝑥.

There are a few ways to approach
this problem. One way would be to find the
equation of the function 𝑔 of 𝑥 given on the graph. As it is a linear function, we know
it can be written in the form 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is the slope and 𝑏
is the 𝑦-intercept. It is clear from the graph that the
𝑦-intercept is negative, four. We know that the slope or gradient
of any line is equal to the rise over the run. Choosing the two points negative
four, four and negative two, zero which lie on the line, the slope is equal to
negative four over two. This is equal to negative two. The function 𝑔 of 𝑥 has equation
negative two 𝑥 minus four. We are told that 𝑔 of 𝑥 is a
reflection of 𝑓 of 𝑥 in the 𝑦-axis. Therefore, 𝑓 of 𝑥 is also a
reflection of 𝑔 of 𝑥 in the 𝑦-axis.

This means that 𝑓 of 𝑥 is equal
to 𝑔 of negative 𝑥. The function 𝑓 of 𝑥 is therefore
equal to negative two multiplied by negative 𝑥 minus four. This simplifies to two 𝑥 minus
four. The original linear function 𝑓 of
𝑥 has equation two 𝑥 minus four. An alternative method would have
been to have sketched the reflection of 𝑔 of 𝑥 on the graph. When reflecting a graph in the
𝑦-axis, we know that the 𝑦-intercept remains the same and the roots change
signs. This means that 𝑓 of 𝑥 will still
intercept the 𝑦-axis at negative four and intercept the 𝑥-axis at positive
two. As the slope or gradient of this
line is two and the 𝑦-intercept is negative four, the equation of the line is two
𝑥 minus four. This confirms the answer we found
using our first method.