
1. Identify the exponential functions
(i) x
e
(ii) e
x
(iii) x
2
(iv) 2x
(v) x
√
3
2. What is the range of the function f(x) = 3x
?
(i) (−∞,∞)
(ii) (−∞, 0) ∪ (0,∞)
(iii) (0,∞)
(iv) [0,∞)
3. Evaluate the expression 
1 +
1
x
x
for the given values of x.
Round your answers to 10 decimal places in questions (i) through (iv),
and give the exact value for question (v).
(i) 10
(ii) 100
(iii) 1000
(iv) 10000
(v) x → ∞
4. The continuous compounding formula is
(i) P ert (ii) P(1 + r
n
)
nt
Where P is the initial invested amount, r the annual interest rate, n the number of
compoundings per year, and t the number of years.

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)**.

---

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.

1. Identify the exponential functions (i) x^e (ii) e^x (iii) x^2 (iv) 2^x (v) x^√3 2. What is the range of the function f(x) = 3^x? (i) (−∞,∞) (ii) (−∞, 0) ∪ (0,∞) (iii) (0,∞) (iv) [0,∞) 3. Evaluate the expression (1 + 1/x)^x for the given values of x. Round your answers to 10 decimal places in questions (i) through (iv), and give the exact value for question (v). (i) 10 (ii) 100 (iii) 1000 (iv) 10000 (v) x → ∞ 4. The continuous compounding formula is (i) P * e^rt (ii) P(1 + r/n)^nt Where P is the initial invested amount, r the annual interest rate, n the number of compoundings per year, and t the number of years.

Explore exponential functions like e^x and 2^x, evaluate the limit of (1 + 1/x)^x as x approaches infinity, and understand the range of f(x) = 3^x. Learn the continuous compounding formula P * e^rt, used in finance to calculate interest over time. Suitable for high school students in Grades 10-12.

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