The question in the image asks: "Which type of function is shown in the table below?" The table lists values of $x$ and $f(x)$. The task is to determine the type of function represented.

Step-by-Step Analysis:

1. Input-Output Relationship (x and f(x)):

   • The table provides these pairs: &x = 2, \, f(x) = 32, \\ &x = 3, \, f(x) = 128, \\ &x = 4, \, f(x) = 256, \\ &x = 5, \, f(x) = 512. \end{aligned}$$

2. Function Check:

   • Each $x$ value has one corresponding $f(x)$. This confirms it is a function.

3. Type of Function:

   • Observing the growth pattern, the values of $f(x)$ multiply by 4 as $x$ increases by 1:
     • $f(3)/f(2) = 128/32 = 4$,
     • $f(4)/f(3) = 256/128 = 4$,
     • $f(5)/f(4) = 512/256 = 4$.
   • The outputs grow exponentially. This suggests an exponential function of the form: $f(x) = a \cdot b^x.$
   • Testing with $f(x) = 2^x \cdot 8$ matches the table. Therefore, it is exponential.

4. Conclusion:

   • The correct answer is: Exponential.

Do you need more details or explanation?

Related Questions:

1. What is the formula of the exponential function represented here?
2. How do you distinguish between linear, quadratic, and exponential functions using a table?
3. Can the base of an exponential function be a number other than 2?
4. What is the significance of the constant ratio property in exponential functions?
5. How would the graph of this function look, and why?

Tip:

When analyzing tables, look for constant differences (linear), constant second differences (quadratic), or constant ratios (exponential).