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Exponential Functions

1️⃣ Understanding Exponential Functions

An exponential function is a function where the variable appears as an exponent. It is written in the form:

\[ f(x)=a\cdot b^x \]

  • ✔ \( a \) is the initial value (the value when \( x=0 \)).
  • ✔ \( b \) is the base (the constant growth or decay factor).
  • ✔ \( x \) is the exponent.

📌 If \( b>1 \), the function represents exponential growth (values increase).
📌 If \( 0exponential decay (values decrease).

Examples:

  • Growth: \( f(x)=3\cdot2^x \) – values double each time.
  • Decay: \( g(x)=100\cdot(0.5)^x \) – values halve each time.

Not Exponential: \( f(x)=x^2 \) (quadratic) and \( f(x)=3x+5 \) (linear).

2️⃣ Key Features of Exponential Functions

  • Initial Value \( a \): The starting value when \( x=0 \).
  • Growth Factor \( b \): Determines how fast the function grows or decays.
  • Horizontal Asymptote: The graph approaches \( y=0 \) but never touches it.
  • Domain and Range:
    • Domain: \( (-\infty, \infty) \).
    • Range: For growth functions, \( (0,\infty) \).

Example: For \( f(x)=5\cdot3^x \), the initial value is 5, the growth factor is 3, and the horizontal asymptote is \( y=0 \).

3️⃣ Exponential Growth and Decay

Exponential functions model growth and decay as follows:

  • Exponential Growth: \( f(x)=a\cdot (1+r)^x \)
  • Exponential Decay: \( f(x)=a\cdot (1-r)^x \)

Growth Example: A population of 1000 growing at 8% per year: \[ P(t)=1000(1.08)^t \] After 5 years, \( P(5)\approx1469 \).

Decay Example: A radioactive substance decaying at 10% per year from 500g: \[ A(t)=500(0.9)^t \] After 3 years, \( A(3)\approx364.5 \) g.

4️⃣ Graphing Exponential Functions

The graph of an exponential function typically has:

  • A horizontal asymptote at \( y=0 \).
  • A curve that rises rapidly (growth) or falls rapidly (decay).
  • A y-intercept at \( (0,a) \).

Example: Graph \( f(x)=2^x \) using the table below:

\( x \) \( f(x)=2^x \)
-2 0.25
-1 0.5
0 1
1 2
2 4

The graph rises quickly for \( x>0 \) and never goes negative.

5️⃣ Real-Life Applications of Exponential Functions

  • Population Growth: A country's population growing at 2% per year.
  • Radioactive Decay: The half-life of substances like carbon-14.
  • Bank Interest: Compound interest follows an exponential model.
  • Virus Spread: Infections can increase exponentially.

Compound Interest Example:

\[ A = P\left(1+\frac{r}{n}\right)^{nt} \]

For a \$1000 investment at 5% annual interest compounded monthly for 10 years,

A = 1000\(\left(1+\frac{0.05}{12}\right)^{12\cdot10}\)

which is approximately \$1647.

🎯 Practice Questions – Fundamental

  1. Identify whether \( f(x)=3^x \), \( g(x)=(0.7)^x \), and \( h(x)=5\cdot2^x \) represent exponential growth or decay.
  2. Find the y-intercept and horizontal asymptote of \( f(x)=4\cdot3^x \).
  3. Solve for \( x \) in \( 2^x=16 \).
  4. A bacteria culture starts with 500 cells and doubles every 3 hours. Find the population after 9 hours.
  5. A car depreciates by 15% per year. If its initial value is \$20,000, what is its value after 5 years?
  6. Graph \( f(x)=5(0.8)^x \).
  7. Calculate the compound interest on a \$2000 investment at 6% annual interest for 8 years, compounded quarterly.
  8. If \( f(x)=10\cdot2^x \), find \( f(3) \).
  9. The half-life of a drug is 6 hours. If 200 mg is taken, how much remains after 18 hours?
  10. Solve: \( 3^{x+1}=81 \).

🔥 Challenging Problems – Push Your Limits!

  1. Solve for \( x \) without logarithms: \( 5^x=125 \).
  2. The population of a city follows \( P(t)=5000(1.04)^t \). Find the time when the population doubles.
  3. A radioactive substance decays according to \( A(t)=100(0.88)^t \). Determine its half-life.
  4. A scientist models virus spread as \( N(t)=10e^{0.3t} \). Find \( N(10) \).
  5. Find the equation of an exponential function passing through \( (0,3) \) and \( (2,12) \).

7️⃣ Summary

  • ✅ Exponential functions are written as \( f(x)=a\cdot b^x \) and represent growth or decay at a constant rate.
  • ✅ Key features include the initial value, growth/decay factor, and a horizontal asymptote at \( y=0 \).
  • ✅ They model real-life phenomena such as population growth, radioactive decay, and compound interest.

8️⃣ Fantastic Job!

Excellent work! Next up: Logarithmic Functions! 🚀

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