We explore what it means to find the zero of a function.
Introduction
In this section, we will explore how to find zeros of more complicated functions.
Finding Zeros of Compositions
Since the outer function is the last function applied, to find the zeros of , we must find the zeros of . Recall that has only one zero, when , since .
We might be tempted to stop here and say that is our zero, but we have to remember we’re working with a composition of functions. We need to find the such that , not the such that . However, the work we’ve done is helpful: we know that plugging 1 into gives 0, so we need to find the numbers we can plug into to get 1, since we’re plugging into . To do this, we set and solve:
Therefore, the zeros of are and .Let’s look closer at what we’ve found. Recall that to be a zero, we need . Let’s check that is in fact a zero. which is what we wanted.
To find the zeros of , we could set and solve, but we’ll show another approach using the composition. First, we find the zeros of the outer function . To do this, we set and solve. This gives us
Next, since , to find the zeros of , we need to find when . This gives us
which is our zero.To find the zeros of compositions of functions, we can find the zeros of the outer function, then, for each zero we’ve found, find the input to the inner function whose output is that zero.