Let's analyze and solve the questions step-by-step.

Question 13: Identify the exponential growth functions.

An exponential growth function is defined as:
$f(x) = a \cdot b^x \quad \text{where} \; b > 1$

Options Analysis:

• A. $f(x) = 2(1.02)^x$:
  Here, $b = 1.02 > 1$, so this is an exponential growth function. ✅

• B. $f(x) = 5000(3)^x$:
  Here, $b = 3 > 1$, so this is also exponential growth. ✅

• C. $f(x) = 75000(0.91)^x$:
  Here, $b = 0.91 < 1$, so this is exponential decay. ❌

• D. $f(x) = 1891(1 - 0.25)^x$:
  Simplifying $b = 1 - 0.25 = 0.75 < 1$, this is also exponential decay. ❌

• E. $f(x) = 2485(1 + 0.25)^x$:
  Simplifying $b = 1 + 0.25 = 1.25 > 1$, this is exponential growth. ✅

Final Answer for Question 13:
The exponential growth functions are: A, B, and E.

Question 14: Identify the correct compound interest model.

The formula for compound interest is:
$A = P \left(1 + \frac{r}{n}\right)^{nt}$

• $P$: Principal amount (initial investment)
• $r$: Annual interest rate
• $n$: Number of compounding periods per year
• $t$: Time in years

Given:

• Annual interest rate $r = 0.03$ (3%)
• Compounded quarterly ($n = 4$)
• Time in years $t = x$

Substitute into the formula:
$A = 150 \left(1 + \frac{0.03}{4}\right)^{4x}$
Simplify:
$A = 150 \left(1 + 0.0075\right)^{4x}$
$A = 150 (1.0075)^{4x}$

Options Analysis:

• A. $150(1 + 0.03)^x$: Incorrect, this assumes annual compounding. ❌
• B. $150(1 + 0.03/4)^x$: Incorrect, the exponent is missing $n = 4$. ❌
• C. $150(1.0075)^{4x}$: Correct, matches the derived formula. ✅
• D. $150(1.03)^{4x}$: Incorrect, the base is wrong. ❌

Final Answer for Question 14:
The correct function is C.

Do you want further explanation or have any doubts?

Related Questions:

1. How can you identify exponential growth versus decay from a function?
2. What is the difference between simple interest and compound interest?
3. How does the frequency of compounding periods affect the growth of an investment?
4. How can the compound interest formula be adjusted for monthly compounding?
5. What happens to the growth rate if the interest rate increases?

Tip: In exponential functions, always check the base $b$. If $b > 1$, it is growth; if $b < 1$, it is decay.