I was wondering whether anyone can explain to me how to go about finding the x intercepts of two functions one being a polynomial and the other a trig function.
For example something like: $f(x) = \cos(\pi x)$ and $g(x)= 4x^2 – 1$.
I need to figure out where they intercept in order to take the integral. I set the two equations equal to each other and move them all to one side, except for the $1$, leaving me with something like $\cos(\pi x) – 4x^2 = -1$.
But I'm not sure where to go from that ( other than desmos.com lol). Any help would be appreciated!
Best Answer
When you have algebraic functions (polynomials) mixed with a transcendental function, there's no systematic way to isolate for the solution of $x$. You're left with having to use numerical methods. For example, if you own a graphing calculator you can graph the two functions and find their intercept.
That being said, in this particular case you can make an observation about when the two functions are equal to zero. In other words, notice that $f(x) = \cos(\pi x)$ is zero at $x = (\frac{1}{2} + k)$ for any $k \in \mathbb{Z}$, and $g(x) = 4x^2 - 1$ is equal to zero at $x = \pm \frac{1}{2}$. Coincidentally, we see here that $x = \pm \frac{1}{2}$ makes both functions equal to zero, and hence equal to each other.
I encourage you to plot the functions to verify that they don't intersect anywhere else (you can use a graphing calculator, wolframalpha.com, MATLAB, whatever you prefer).