To find the exponential regression function of the form $f(x) = ab^x$ through the given points $(3, 2), (4, 4), (5, 11)$, we need to use a regression technique that fits an exponential curve to these points.

Here’s how you can calculate the exponential regression function step by step:

1. Transform the problem:

   • For an exponential model $y = ab^x$, take the natural logarithm on both sides: $\ln(y) = \ln(a) + x \ln(b)$
   • This converts the problem to a linear regression form.

2. Input the points into regression technology:

   • Use software like a graphing calculator, Excel, or online tools like Desmos or GeoGebra, where you can input the points to perform exponential regression directly.

3. Performing the calculation:

   • The exact result will provide the values of $a$ and $b$, which we will round to four decimal places.

Let me perform the calculation for you.The exponential regression function through the given points is:

$f(x) = 0.0981 \times (2.5675)^x$

Would you like more details on how this regression was calculated, or do you have any other questions?

Here are some related questions to consider:

1. What is the difference between linear and exponential regression?
2. How can you use exponential regression to model real-world data?
3. What are some common applications of exponential functions in science and economics?
4. How does the accuracy of the regression change when using more data points?
5. Can you use logarithms to transform other types of models for regression?

Tip: When fitting data with exponential growth or decay, always plot the data first to confirm that an exponential model is appropriate.