To find the point(s) of intersection between the two functions, we need to set them equal to each other, as the intersection occurs where both functions have the same value for the same $x$.

Step 1: Set the equations equal

Given the two functions:

1. $y = 0.19(10^{x}+1)$
2. $y = 6\cos(x + 4.51) - 2.2$

We set them equal to each other: $0.19(10^{x} + 1) = 6\cos(x + 4.51) - 2.2$

Step 2: Solve the equation for $x$

We now have a transcendental equation (an equation involving both algebraic and trigonometric expressions). This is not easy to solve algebraically, but it can be solved using numerical methods such as graphing or using iterative techniques.

Step 3: Graph the functions

We can graph both functions and visually determine the points of intersection, or we can use numerical methods like Newton's method or a graphing calculator to find the roots.

Alternatively, if you have access to a graphing tool, you can plot the two functions and identify where they intersect.

Step 4: Estimate or compute the roots

You can use a numerical solver or tools like WolframAlpha, a graphing calculator, or software like Python (with libraries such as NumPy and SciPy) to find the values of $x$ at which these two functions are equal.

Would you like to go through solving this numerically or graphically in more detail, or would you prefer me to assist with a specific method (like using a calculator or software)?