Quiz on Basic Exponential Functions: Test Your Knowledge

📚 Quick Study Guide: Basic Exponential Functions

• 💡 Definition: An exponential function is a mathematical function of the form $f(x) = ab^x$, where $a$ is a non-zero real number, $b$ is a positive real number not equal to 1, and $x$ is any real number (the exponent).
• 🔢 Components:
  • 📈 Base ($b$): The constant being raised to a power. It determines the growth or decay of the function.
  • 🧪 Exponent ($x$): The variable that determines the power to which the base is raised.
  • 💰 Initial Value ($a$): The value of the function when $x=0$, i.e., $f(0) = a$.
• 📊 Graphs & Characteristics:
  • 📈 If $b > 1$, the function represents exponential growth (graph increases from left to right).
  • 📉 If $0 < b < 1$, the function represents exponential decay (graph decreases from left to right).
  • 🌍 The domain of an exponential function is all real numbers ($\mathbb{R}$).
  • 🎯 The range is all positive real numbers ($ (0, \infty) $) if $a > 0$, or all negative real numbers ($ (-\infty, 0) $) if $a < 0$.
  • 📏 Exponential functions have a horizontal asymptote at $y=0$ (the x-axis) unless shifted vertically.
  • 🔑 They always pass through the point $(0, a)$.
• ✨ Special Base: Euler's Number ($e$):
  • 🌿 The number $e \approx 2.71828$ is a fundamental mathematical constant, often called the natural base.
  • 🚀 Functions with base $e$ are called natural exponential functions, typically written as $f(x) = ae^x$. They are crucial in calculus, finance, and natural sciences.
• 📜 Key Exponent Rules (for any positive base $b$ and real numbers $m, n$):
  • ✖️ Product Rule: $b^m \cdot b^n = b^{m+n}$
  • ➗ Quotient Rule: $\frac{b^m}{b^n} = b^{m-n}$
  • 📈 Power Rule: $(b^m)^n = b^{mn}$
  • 🔄 Negative Exponent Rule: $b^{-n} = \frac{1}{b^n}$
  • 1️⃣ Zero Exponent Rule: $b^0 = 1$ (for $b \neq 0$)

🧠 Practice Quiz: Test Your Knowledge on Exponential Functions

1. Which of the following equations represents an exponential function?

   A. $f(x) = x^2 + 3x - 1$
   B. $f(x) = 5x + 2$
   C. $f(x) = 3^x$
   D. $f(x) = \sqrt{x} + 4$

2. If $f(x) = 2 \cdot 3^x$, what is the value of $f(2)$?

   A. 12
   B. 18
   C. 8
   D. 36

3. An exponential function $f(x) = ab^x$ represents decay if:

   A. $b > 1$
   B. $b = 1$
   C. $0 < b < 1$
   D. $b < 0$

4. What is the horizontal asymptote of the function $f(x) = 4^x$?

   A. $x = 0$
   B. $y = 0$
   C. $y = 1$
   D. There is no horizontal asymptote.

5. Simplify the expression: $(2^3)^2$

   A. $2^5$
   B. $2^6$
   C. $4^3$
   D. $2^9$

6. What is the value of $5 \cdot e^0$?

   A. 0
   B. 1
   C. 5
   D. $e$

7. Which point does the graph of $f(x) = 3 \cdot 2^x$ always pass through?

   A. $(0, 2)$
   B. $(0, 3)$
   C. $(1, 6)$
   D. $(3, 0)$