Properties of exponential functions

Properties of Exponential Functions

An exponential function is a mathematical function of the form f(x) = a × b^x, where:

• a is a constant (the initial value),
• b is the base of the exponential (a positive real number),
• x is the exponent (the variable).

The base b determines the behaviour of the function. If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing.

Key Properties of Exponential Functions

Exponential functions have several important properties:

1. Domain and Range: The domain of an exponential function is all real numbers (−∞, ∞). The range is (0, ∞) if a > 0, and (−∞, 0) if a < 0.
2. Intercepts: The y-intercept occurs at (0, a). There are no x-intercepts for a > 0. If a < 0, the function does not cross the x-axis.
3. Asymptotes: The horizontal asymptote is the line y = 0. The function approaches this line but never touches it.
4. Growth and Decay: If b > 1, the function represents exponential growth. If 0 < b < 1, it represents exponential decay.

Examples of Exponential Functions

Example 1: Exponential Growth

Consider the function f(x) = 2 × 3^x. Here, a = 2 and b = 3.

1. The domain is all real numbers: (−∞, ∞).
2. The range is (0, ∞) because a > 0.
3. The y-intercept is at (0, 2) since f(0) = 2 × 3^0 = 2.
4. As x increases, f(x) grows rapidly due to the base 3.

Example 2: Exponential Decay

Now, consider the function g(x) = 5 × (0.5)^x. Here, a = 5 and b = 0.5.

1. The domain is all real numbers: (−∞, ∞).
2. The range is (0, ∞) since a > 0.
3. The y-intercept is at (0, 5) since g(0) = 5 × (0.5)^0 = 5.
4. As x increases, g(x) decreases towards 0.

Transformations of Exponential Functions

Exponential functions can be transformed by shifting, reflecting, or stretching them. These transformations include:

• Vertical Shift: f(x) = a × b^x + k shifts the graph up (if k > 0) or down (if k < 0).
• Horizontal Shift: f(x) = a × b^(x − h) shifts the graph right (if h > 0) or left (if h < 0).
• Reflection: f(x) = −a × b^x reflects the graph over the x-axis.
• Stretching and Compressing: f(x) = c × a × b^x, where c > 1 stretches the graph vertically and 0 < c < 1 compresses it.

Example 3: Vertical Shift

Consider the function h(x) = 2 × 3^x + 1. This function has a vertical shift of 1 unit upwards.

1. The y-intercept is now at (0, 3) since h(0) = 2 × 3^0 + 1 = 3.
2. The horizontal asymptote is now at y = 1.

Example 4: Horizontal Shift

For the function k(x) = 2 × 3^(x − 2), there is a horizontal shift of 2 units to the right.

1. The y-intercept occurs at (2, 2) since k(2) = 2 × 3^(2 − 2) = 2.
2. The horizontal asymptote remains at y = 0.

Applications of Exponential Functions

Exponential functions are used in various fields, such as finance, biology, and physics. They can model growth processes, such as population growth, and decay processes, such as radioactive decay.

Example 5: Population Growth

A population of bacteria doubles every 3 hours. If the initial population is 100, the function can be expressed as:

P(t) = 100 × 2^(t/3)

where P(t) is the population at time t (in hours).

Example 6: Radioactive Decay

A certain substance has a half-life of 5 years. If the initial amount is 80 grams, the amount remaining after t years can be expressed as:

A(t) = 80 × (0.5)^(t/5)

where A(t) is the amount remaining after time t.

Solving Exponential Equations

To solve equations involving exponential functions, you can use logarithms. Logarithms are the inverse operations of exponentiation.

Example 7: Solving an Exponential Equation

Consider the equation 3^x = 81.

1. Rewrite 81 as a power of 3: 81 = 3^4.
2. This gives us the equation: 3^x = 3^4.
3. Since the bases are the same, set the exponents equal: x = 4.

Example 8: Using Logarithms to Solve

Now consider the equation 2^x = 10.

1. Take the logarithm of both sides: log(2^x) = log(10).
2. Use the power rule of logarithms: x × log(2) = log(10).
3. Now solve for x: x = log(10) / log(2).

Summary

• Exponential functions have specific properties such as domain, range, intercepts, and asymptotes.
• Transformations can shift, reflect, or stretch the graph of exponential functions.
• Exponential functions are used in real-world applications like population growth and radioactive decay.
• Logarithms can be used to solve exponential equations.

Check your understanding

• What is the domain and range of the function f(x) = 4 × 2^x?
• How does the graph of g(x) = 3 × (0.75)^x differ from the graph of f(x) = 3 × 2^x?
• What is the solution to the equation 5^x = 125?
• Explain how to find the horizontal asymptote of an exponential function.