Let's analyze and solve the problems step by step from the images.

Part (a) - Is $a > 1$ or $0 < a < 1$?

From the graph provided, the exponential curve is decreasing. For an exponential function $f(x) = a^x$, if $a > 1$, the function is increasing. If $0 < a < 1$, the function is decreasing. Since the graph shows a decreasing function, we can conclude that:

• $0 < a < 1$.

Part (b) - Domain and range of $f(x) = a^x$

For any exponential function $f(x) = a^x$, regardless of the value of $a$:

• The domain is all real numbers: $(-\infty, \infty)$.
• The range is $(0, \infty)$, because exponential functions never touch or go below the x-axis.

So, the answers are:

• Domain: $(-\infty, \infty)$
• Range: $(0, \infty)$

Part (c) - Sketch the graph of $g(x) = a^x + 2$

This graph will be the same as the original exponential graph $f(x) = a^x$, but shifted upward by 2 units. The entire graph will move up, so the horizontal asymptote will now be at $y = 2$ instead of $y = 0$.

From the given options, the correct graph would be option B, which shows an exponential curve shifted upward.

Part (d) - Domain and range of $g(x) = a^x + 2$

The domain of $g(x) = a^x + 2$ remains the same as $f(x) = a^x$:

• Domain: $(-\infty, \infty)$.

However, the range changes because the graph is shifted upwards by 2 units:

• The new range is $(2, \infty)$, since the curve never goes below $y = 2$.

Part (e) - Sketch the graph of $h(x) = a^{x + 3}$

The function $h(x) = a^{x+3}$ represents a horizontal shift of the graph of $f(x) = a^x$ 3 units to the left.

From the given options, the correct graph would be option C, which shows a horizontal shift of the curve to the left.

Part (f) - Domain and range of $h(x) = a^{x + 3}$

The domain of $h(x) = a^{x+3}$ remains unchanged:

• Domain: $(-\infty, \infty)$.

The range also remains the same as the original exponential function, since a horizontal shift does not affect the range:

• Range: $(0, \infty)$.

Do you want further explanations on any part? Here are 5 related questions to further expand on this topic:

1. How do vertical shifts affect the range of an exponential function?
2. What happens to the graph of $f(x) = a^x$ when $a = 1$?
3. How would a horizontal shift of $f(x) = a^x$ to the right by 5 units be written algebraically?
4. What is the impact on the graph of $f(x) = a^x$ when you reflect it over the y-axis?
5. Can the range of an exponential function ever be negative?

Tip: To sketch transformations, always think about how horizontal and vertical shifts affect the asymptotes and key points on the original graph.