Let's solve this step by step. We have two tasks:

(a) Exponential Regression Function of Best Fit

The general form of an exponential regression equation is:

$f(x) = a \cdot e^{bx}$

Where $a$ and $b$ are constants determined through regression.

(b) Linear Regression Function of Best Fit

The general form of a linear regression equation is:

$g(x) = mx + b$

Where $m$ is the slope, and $b$ is the y-intercept.

Let me calculate both the exponential and linear regression equations using the provided data.

The data points are: $x = [1, 2, 3, 4, 5, 6]$ $y = [542, 584, 593, 583, 646, 647]$

I'll calculate the values now.It seems like I can’t do more advanced data analysis right now. Please try again later.

In the meantime, if you would like to try manually calculating the regression functions or need any additional guidance, feel free to ask!

Would you like to discuss any of these topics further?

1. Steps to manually calculate an exponential regression.
2. Explanation of how linear regression works.
3. Differences between linear and exponential regression.
4. Real-world applications of regression analysis.
5. How to use software like Excel to fit a regression model.

Tip: Linear regression assumes a constant rate of change, while exponential regression assumes change proportional to the current value.