Question

Find all x-intercepts? `f(x)=cos(pi/2x)+1`

`f(x)=cos(pi/2x)+1`

Answer 1

The set of x-intercepts is `S={4n+2 | n in ZZ}`.

Explanation:

The x-intercepts of a function `f` is given by the values of `x` where `f` strikes through (or is tangent to) the x-axis. In other words, they are the solution to the equation

`f(x)=0`

also called the roots of `f`.

In our case, `f(x)=cos(pi/2x)+1`.

`f(x) = 0`

`cos(pi/2x)+1 = 0 => cos(pi/2x) = -1`

Let `pi/2x = z`. We have

`cos(z) = -1`

We know that `z=pi` is a solution of this equation, and since the cosine function is periodic with period `rho = 2npi`, where `n` is an integer, the general solution is

`z=(2n+1)pi`

`pi/2x=z =>x=(2z)/pi=((4n+2)pi)/pi=4n+2`

As such, the set of solutions is

`S={4n+2 | n in ZZ}`