Let's break down each question to provide answers and explanations.

1. Identify the exponential functions

Exponential functions are those of the form $f(x) = a \cdot e^{bx}$ or $f(x) = a \cdot b^x$, where $e$ or any constant is raised to a variable exponent.

(i) $x^e$: Not exponential (This is a power function where a variable is raised to a constant power.) (ii) $e^x$: Exponential (The base $e$ is raised to the variable $x$.) (iii) $x^2$: Not exponential (Again, this is a power function.) (iv) $2^x$: Exponential (The base 2 is raised to the variable $x$.) (v) $x^{\sqrt{3}}$: Not exponential (Another power function.)

So, the exponential functions are:

• (ii) $e^x$
• (iv) $2^x$

2. What is the range of the function $f(x) = 3^x$?

The function $f(x) = 3^x$ is an exponential function where the base $3$ is greater than 1. For such functions:

• As $x$ approaches negative infinity, $f(x)$ approaches 0 but never reaches it.
• As $x$ approaches positive infinity, $f(x)$ increases without bound.

So, the range is $(0, \infty)$.

Correct answer: (iii) (0, \infty)

3. Evaluate the expression $\left(1 + \frac{1}{x}\right)^x$ for the given values of $x$.

This expression approaches the mathematical constant $e$ as $x$ increases.

Let's calculate for each value:

(i) For $x = 10$: $\left(1 + \frac{1}{10}\right)^{10} = \left(1.1\right)^{10} \approx 2.5937424601$

(ii) For $x = 100$: $\left(1 + \frac{1}{100}\right)^{100} = \left(1.01\right)^{100} \approx 2.7048138294$

(iii) For $x = 1000$: $\left(1 + \frac{1}{1000}\right)^{1000} \approx 2.7169239322$

(iv) For $x = 10000$: $\left(1 + \frac{1}{10000}\right)^{10000} \approx 2.7181459268$

(v) As $x$ approaches infinity, $\left(1 + \frac{1}{x}\right)^x$ approaches the exact value $e$: $e \approx 2.7182818285$

4. The continuous compounding formula

The continuous compounding formula is given by: $\text{(i)}\ P \cdot e^{rt}$ Where:

• $P$ is the initial invested amount,
• $r$ is the annual interest rate,
• $t$ is the number of years.

The second option, $P(1 + \frac{r}{n})^{nt}$, is the formula for compound interest, not continuous compounding.

So, the correct formula for continuous compounding is (i).

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Would you like more details on any of these solutions? Here are five questions to explore these topics further:

1. What are the differences between exponential and power functions?
2. How do you determine the range of an exponential function with different bases?
3. Why does the expression $\left(1 + \frac{1}{x}\right)^x$ approach $e$ as $x$ increases?
4. How does the compound interest formula change when the interest is compounded quarterly, monthly, or daily?
5. What are some real-world applications of continuous compounding?

Tip: When working with exponential functions, always pay attention to the base, as it determines the behavior of the function over its domain.