Given the function f(x) = sin(x) and g(x) = ax
We need to find the points of intersection of f(x) and g(x).
The points of intersections are x an y values such that f(x) = g(x).
==> sin(x) = ax.
We know that if sin(x) = x, then one answer is x= 0.
==> sin0 = 0
==> Then the intersection point is the point ( 0, 0) where a could be any number.
However, when using a graph, we notice that there is another intersection point that we can only obtain by graphing and estimating the value.
First, we know that sin(x) belongs to the interval [-1,1].
Then, -1 =< sinx =< 1
Then -1 =< ax =< 1
==> -1/a =< x =< 1/a
There is no exact solution or way find the other intersection point but estimating using the graph and a calculator.
At the point of intersection of f(x) and g(x), both of the functions have the same value.
So f(x) = g(x)
=> sin x = ax
Now, we see that sin x cannot be equal to ax unless x = 0, when sin x = 0 and ax = 0. For any other value of x, sin x is not equal to ax.
Therefore the required point of intersection is at x = 0.