Exponential Functions

Introduction to Exponential Functions

Exponential functions are a fundamental class of functions in mathematics, widely used to model growth and decay processes in various fields such as finance, biology, physics, and economics. An exponential function is typically written in the form f(x) = b^x, where the base b is a positive real number not equal to 1.

• Definition: An exponential function is a function of the form f(x) = b^x, where b > 0 and b ≠ 1.

• Domain: The domain of an exponential function is all real numbers: {x : }.

• Range: The range is all positive real numbers: {y : }.

• Key Property: For all b > 0, , so f(0) = 1.

• Monotonicity: If b > 1, the function is increasing; if 0 < b < 1, the function is decreasing.

Exponent Rules

Exponential expressions follow several important algebraic rules, which are essential for simplifying and manipulating exponential functions.

• Product Rule:

• Quotient Rule:

• Power Rule:

• Zero Exponent: for all

Evaluating Exponential Functions

Exponential functions can be evaluated for integer, rational, and irrational exponents:

• Integer Exponents:

• Negative Integer Exponents:

• Rational Exponents:

• Irrational Exponents: For example, is defined and can be approximated using rational exponents close to .

Additional info: The domain of exponential functions is all real numbers, and their range is all positive real numbers. The function is continuous and smooth for all real .

Properties of Exponential Functions

• Domain:

• Range:

• Intercept:

• Growth Rate: Larger values of produce greater rates of increase in if .

• Monotonicity: If , increases as increases; if , decreases as increases.

Example: Comparing Exponential Growth

• Let and . For , grows faster than because the base is larger.

• For , and .

Inverse Functions

Every exponential function has an inverse, which is a logarithmic function. The logarithmic function is discussed in detail in later sections.