1. The problem states that the function $f(x) = \cos(x)$ is transformed by stretching it horizontally by a factor of 4 and shifting it 1 unit up. 2. The formula for horizontal stretching by a factor of $k$ is $f(x) \to f\left(\frac{x}{k}\right)$. 3. The formula for vertical shifting up by $c$ units is $f(x) \to f(x) + c$. 4. Applying the horizontal stretch by 4 to $f(x) = \cos(x)$ gives: $$g(x) = \cos\left(\frac{x}{4}\right)$$ 5. Then shifting this function 1 unit up results in: $$g(x) = \cos\left(\frac{x}{4}\right) + 1$$ 6. Therefore, the equation of the new function is: $$g(x) = \cos\left(\frac{x}{4}\right) + 1$$