The activity requires finding the domain and range of various exponential functions. Let’s analyze each function:

1. $f(x) = \left( \frac{1}{2} \right)^{x+1}$

   • Domain: Exponential functions are defined for all real numbers.
     • Domain: $(-\infty, \infty)$
   • Range: Since the base $\frac{1}{2}$ is positive and the function is decreasing (less than 1), the range will be positive values.
     • Range: $(0, \infty)$

2. $f(x) = 3^{x+2}$

   • Domain: $(-\infty, \infty)$
   • Range: Since the base is 3 (greater than 1), this is a typical exponential growth function.
     • Range: $(0, \infty)$

3. $f(x) = -(5)^x$

   • Domain: $(-\infty, \infty)$
   • Range: The function is the negative of an exponential growth function, so it will be negative.
     • Range: $(-\infty, 0)$

4. $f(x) = 4^{-x}$

   • Domain: $(-\infty, \infty)$
   • Range: $4^{-x}$ is still a positive decreasing function.
     • Range: $(0, \infty)$

5. $f(x) = \left( \frac{1}{3} \right)^{-x+1}$

   • Domain: $(-\infty, \infty)$
   • Range: In this case, the base $\frac{1}{3}$ flips due to the negative exponent, so this is an increasing exponential function.
     • Range: $(0, \infty)$

6. $f(x) = 2^{x-1} + 1$

   • Domain: $(-\infty, \infty)$
   • Range: The shift by +1 moves the range up, so it will never reach 1.
     • Range: $(1, \infty)$

7. $f(x) = -5^x - 2$

   • Domain: $(-\infty, \infty)$
   • Range: Since it's the negative of an exponential and shifted by -2, the values will always be less than -2.
     • Range: $(-\infty, -2)$

8. $f(x) = \left( \frac{1}{4} \right)^{2x+1} + 3$

   • Domain: $(-\infty, \infty)$
   • Range: Since the base is less than 1 and shifted up by 3, the function never reaches 3.
     • Range: $(3, \infty)$

9. $f(x) = 5^{3x} - 4$

   • Domain: $(-\infty, \infty)$
   • Range: A standard growth exponential function minus 4, so the minimum is -4.
     • Range: $(-4, \infty)$

10. $f(x) = -\left( \frac{2}{3} \right)^x - 3$

    • Domain: $(-\infty, \infty)$
    • Range: Since it's the negative of a decaying exponential function, shifted down by -3, the range is below -3.
      • Range: $(-\infty, -3)$

Do you need detailed steps on any of these, or have any specific questions?

Here are 5 related questions to explore further:

1. How do vertical shifts affect the range of exponential functions?
2. What happens to the range of an exponential function when the base is a fraction?
3. How does a negative sign outside the exponential affect the function’s behavior?
4. What is the general form of the domain for any exponential function?
5. How can the concept of horizontal shifts affect the graph of exponential functions?

Tip: Always pay attention to transformations (shifts, reflections) when finding the range of exponential functions!