Here is the breakdown and solution for the task provided:

Problem Statement:

The function is given as $f(x) = 34 \cdot 1.05^x$. There are multiple subquestions:

a) Explain the significance of the numbers 34 and 1.05 in the function.

• 34: This is the initial value of the function, $f(0)$, or the starting amount when $x = 0$.
• 1.05: This represents the growth factor. A base greater than 1 (like 1.05) indicates exponential growth, with a 5% increase per unit increase in $x$.

b) Is the function increasing, decreasing, or neither? Justify your answer.

The function $f(x) = 34 \cdot 1.05^x$ is increasing, because the growth factor $1.05 > 1$. This ensures that as $x$ increases, $f(x)$ also increases exponentially.

c) Calculate $f(0)$, $f(20)$, and $f(100)$.

Using $f(x) = 34 \cdot 1.05^x$:

1. $f(0) = 34 \cdot 1.05^0 = 34 \cdot 1 = 34$
2. $f(20) = 34 \cdot 1.05^{20}$
3. $f(100) = 34 \cdot 1.05^{100}$

Let’s calculate $f(20)$ and $f(100)$:

• $f(20) = 34 \cdot 2.6533 = 90.1922$ (approximately)
• $f(100) = 34 \cdot 131.501 = 4,471.03$ (approximately)

d) Draw the graph of $f(x)$ from $x = 0$ to $x = 100$.

I'll provide a graph below using computational tools. numpy as np import matplotlib.pyplot as plt

Define the function

def f(x): return 34 * (1.05 ** x)

Define the range for x values

x = np.linspace(0, 100, 500) y = f(x)

Plot the graph

plt.figure(figsize=(10, 6)) plt.plot(x, y, label=r"$f(x) = 34 \cdot 1.05^x$", color="blue") plt.title("Graph of f(x) = 34 · 1.05^x", fontsize=14) plt.xlabel("x", fontsize=12) plt.ylabel("f(x)", fontsize=12) plt.grid() plt.legend() plt.xlim(0, 100) plt.ylim(0, 5000) plt.show()Here is the graph of $f(x) = 34 \cdot 1.05^x$ for $x$ in the range [0, 100]. The function grows exponentially, with rapid growth as $x$ increases.

e) Solve the equation $f(x) = 1000$.

We solve $34 \cdot 1.05^x = 1000$:

$1.05^x = \frac{1000}{34} = 29.4118$

Take the natural logarithm on both sides:

$x \ln(1.05) = \ln(29.4118)$

$x = \frac{\ln(29.4118)}{\ln(1.05)}$

Let’s calculate this value: math

Calculate x

numerator = math.log(1000 / 34) denominator = math.log(1.05) x_solution = numerator / denominator x_solutionThe solution to the equation $f(x) = 1000$ is approximately $x = 69.30$.

f) Explain how the solution can be found on the graph.

To find the solution graphically, locate $f(x) = 1000$ on the y-axis. Then, draw a horizontal line from this point until it intersects the curve of $f(x)$. The x-coordinate of this intersection point corresponds to the solution $x \approx 69.3$.

Do you want further clarification or details on any of these subquestions?
Here are 5 related questions to expand your understanding:

1. What would happen to the function if the growth factor were less than 1, e.g., $1.05$ replaced by $0.95$?
2. How would the graph change if the initial value $34$ was doubled?
3. Can you derive the general formula for $x$ when solving $f(x) = C$?
4. What does the derivative of $f(x)$ tell you about the rate of growth of the function?
5. How can the exponential function be applied to real-world problems such as population growth or investments?

Tip: Always check for logarithmic solutions when solving exponential equations—it simplifies the process significantly.