# How do you find the point of intersection between f(x)=sin(x) and g(x)=ax?

hala718 | Certified Educator

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.

justaguide | Certified Educator

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.