Example 1: evaluating the function

1. If f(x) = 2^x, find f(3) and f(−2).
2. f(3) = 2³ = 8. And f(−2) = 2⁻² = 1 / 2² = 1/4.

Example 2: compound growth in a bank

• A person deposits 1000 Egyptian pounds in a bank that pays 5% interest compounded yearly. What is the amount after 3 years?
• Solution A = 1000 × (1 + 0.05)³ = 1000 × 1.157625 = 1157.625, which is about 1157.63 Egyptian pounds.

Example 3: half-life

• A radioactive substance has a mass of 100 grams and a half-life of 5 years. What mass remains after 10 years, and after 3 years?
• Solution After 10 years: N = 100 × (1/2)^10/5 = 100 × (1/2)² = 25 grams. After 3 years: N = 100 × (1/2)^3/5 ≈ 66 grams (about 65.98 grams).

Example 4: depreciation of a car

• A car is bought for 20000 Egyptian pounds and loses 15% of its value each year. What is its value after 2 years?
• Solution Each year the car keeps 1 − 0.15 = 0.85 of its value. So V = 20000 × (0.85)² = 20000 × 0.7225 = 14450 Egyptian pounds.

Example 5: finding the base from a point

Example 6: growth of bacteria

• A culture starts with 50 bacteria and doubles every 3 hours. Write the rule of the function, then find the number after 12 hours.