Introduction to Exponential Functions

Overview

Exponential functions are mathematical functions of the form:

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

Where:

  • \(a\): The initial value (or y-intercept when \(x = 0\))
  • \(b\): The base of the exponential function
  • \(x\): The exponent (independent variable)

Exponential functions are widely used in modeling growth and decay, such as population growth, radioactive decay, and compound interest.

Characteristics of Exponential Functions

  • When \(b > 1\), the function exhibits exponential growth.
  • When \(0 < b < 1\), the function exhibits exponential decay.
  • The graph has a horizontal asymptote at \(y = 0\).

Example: Exponential Growth

Consider the exponential function:

\( f(x) = 2 \cdot 3^x \)

This represents exponential growth because the base \(b = 3 > 1\). The graph of this function increases rapidly as \(x\) increases.

Example: Exponential Decay

Consider the exponential function:

\( f(x) = 5 \cdot (0.5)^x \)

This represents exponential decay because the base \(b = 0.5 < 1\). The graph of this function decreases rapidly as \(x\) increases.

Practice Questions

  1. Question 1: Identify the type of exponential function (growth or decay) and describe the behavior of the graph.

    \( f(x) = 4 \cdot 1.2^x \)

    Solution

    This is exponential growth because the base \(b = 1.2 > 1\). The graph increases rapidly as \(x\) increases.

  2. Question 2: Write the equation of an exponential function where the initial value is 3, and the base is 0.75.
    Solution

    The equation is:

    \( f(x) = 3 \cdot (0.75)^x \)

    This represents exponential decay because \(b = 0.75 < 1\).

  3. Question 3: Evaluate the exponential function at \(x = 2\):

    \( f(x) = 2 \cdot 5^x \)

    Solution

    Substitute \(x = 2\):

    \( f(2) = 2 \cdot 5^2 = 2 \cdot 25 = 50 \)

    The value of the function is \(50\).

  4. Question 4: Match the following equations to their descriptions (exponential growth or decay):
    • (a) \( f(x) = 6 \cdot 1.5^x \)
    • (b) \( f(x) = 4 \cdot (0.3)^x \)
    Solution

    (a) Exponential growth (\(b = 1.5 > 1\))

    (b) Exponential decay (\(b = 0.3 < 1\))

  5. Question 5: Sketch the graph of \( f(x) = 3 \cdot 2^x \). Identify the y-intercept and the horizontal asymptote.
    Solution

    The y-intercept is at \(y = 3\) (when \(x = 0\)).

    The horizontal asymptote is at \(y = 0\).

→Properties of Exponential Functions