Let's tackle Question 1 step-by-step:

Part (a)

1. i) Sketch the curve $y = 2^x$ for $-1 \leq x \leq 3$:

   • The function $y = 2^x$ is an exponential curve. To sketch it over the interval $-1 \leq x \leq 3$, let's evaluate some key points:
     • $x = -1$: $y = 2^{-1} = \frac{1}{2} = 0.5$
     • $x = 0$: $y = 2^0 = 1$
     • $x = 1$: $y = 2^1 = 2$
     • $x = 2$: $y = 2^2 = 4$
     • $x = 3$: $y = 2^3 = 8$
   • The curve will rise exponentially as $x$ increases, starting from a small value at $x = -1$ and quickly increasing towards $y = 8$ at $x = 3$.

2. ii) What value does $y$ approach as $x$ gets smaller?

   • As $x$ approaches negative infinity, $y = 2^x$ approaches 0, but it never actually reaches 0.

Would you like a detailed sketch of the graph or further explanations?

Here are some additional questions for exploration:

1. How does changing the base of the exponential function (e.g., using 3 instead of 2) affect the curve?
2. What is the effect on the graph if we add a constant, such as $y = 2^x + 1$?
3. How would the graph change if we had $y = 2^{-x}$?
4. How can we determine the horizontal asymptote of an exponential function?
5. How does the function behave for non-integer values of $x$?

Tip: In exponential functions, the base determines the growth rate—larger bases result in faster growth.