Properties of Exponential Functions
Overview
Exponential functions have several key properties that define their behavior. An exponential function has the form:
\( f(x) = a \cdot b^x \)
Where \(a\) is the initial value, \(b\) is the base, and \(x\) is the exponent. The properties of these functions depend heavily on the value of the base \(b\).
Key Properties of Exponential Functions
- Domain: The domain of exponential functions is all real numbers (\(-\infty, \infty\)).
- Range: The range is always positive for all exponential functions, meaning \(f(x) > 0\) for all values of \(x\).
- Asymptote: Exponential functions have a horizontal asymptote at \(y = 0\), which the graph approaches but never touches.
- Increasing or Decreasing: If \(b > 1\), the function exhibits exponential growth. If \(0 < b < 1\), the function exhibits exponential decay.
- Intercept: The y-intercept of an exponential function is the value of \(f(0)\), which equals \(a\). This is the starting point of the function when \(x = 0\).
- End Behavior: As \(x \to \infty\), \(f(x)\) approaches infinity if \(b > 1\), or it approaches 0 if \(0 < b < 1\).
Example 1: Exponential Growth
Consider the function:
\( f(x) = 2 \cdot 3^x \)
This is an example of exponential growth because \(b = 3 > 1\). The graph will increase rapidly as \(x\) increases, and it will approach the horizontal asymptote \(y = 0\) as \(x \to -\infty\). The y-intercept is at \(f(0) = 2\).
Example 2: Exponential Decay
Consider the function:
\( f(x) = 5 \cdot (0.5)^x \)
This is an example of exponential decay because \(b = 0.5 < 1\). The graph will decrease rapidly as \(x\) increases, and it will approach the horizontal asymptote \(y = 0\) as \(x \to \infty\). The y-intercept is at \(f(0) = 5\).
Practice Questions
- Question 1: Identify the properties of the function \(f(x) = 3 \cdot 2^x\). Is it an example of exponential growth or decay? What is the y-intercept, and what is the horizontal asymptote?
Solution
The function is an example of exponential growth because \(b = 2 > 1\). The y-intercept is at \(f(0) = 3\), and the horizontal asymptote is at \(y = 0\).
- Question 2: For the function \(f(x) = 4 \cdot (0.25)^x\), what is the end behavior as \(x \to \infty\)? Is the function increasing or decreasing?
Solution
The function is decreasing because \(b = 0.25 < 1\). As \(x \to \infty\), \(f(x) \to 0\) (approaches the horizontal asymptote at \(y = 0\)).
- Question 3: Write the equation of an exponential function with the following properties:
- The function has a y-intercept of 8.
- The function is decreasing.
- The base is 0.4.
Solution
The equation of the exponential function is:
\( f(x) = 8 \cdot (0.4)^x \)
- Question 4: Given the exponential function \(f(x) = 10 \cdot (1.5)^x\), what is the range of the function?
Solution
The range of the function is all positive real numbers: \(f(x) > 0\) for all \(x\).
- Question 5: Determine the horizontal asymptote and end behavior for the function \(f(x) = 3 \cdot 0.2^x\).
Solution
The horizontal asymptote is at \(y = 0\), and as \(x \to \infty\), \(f(x) \to 0\) (the function decays towards 0).
- Question 6: Sketch the graph of the function \(f(x) = 1 \cdot 5^x\) and identify the following:
- The y-intercept.
- The horizontal asymptote.
- The end behavior as \(x \to -\infty\).
Solution
The y-intercept is at \(f(0) = 1\), the horizontal asymptote is at \(y = 0\), and as \(x \to -\infty\), \(f(x) \to 0\).