FUNÇÃO EXPONENCIAL - DEFINIÇÃO, VALOR NUMÉRICO, PROPRIEDADES E GRÁFICOS

Key Concepts

• Exponential Function: A function of the form f(x) = a^x, where 'a' is a constant base and 'x' is the variable in the exponent.
• Base (a): The constant 'a' in the exponential function a^x. It must be greater than zero and different from one (a > 0 and a ≠ 1).
• Exponent (x): The variable 'x' in the exponential function a^x. It can be any real number.
• Increasing Exponential Function: An exponential function where the base 'a' is greater than one (a > 1). As x increases, f(x) also increases.
• Decreasing Exponential Function: An exponential function where the base 'a' is between zero and one (0 < a < 1). As x increases, f(x) decreases.
• Numerical Value of an Exponential Function: The value of the function f(x) for a specific value of x.
• Exponential Equation: An equation where the variable appears in the exponent.
• Domain: The set of all possible input values (x-values) for which the function is defined.
• Range: The set of all possible output values (y-values or f(x)-values) that the function can produce.
• Injective Function: A function where each input value (x) maps to a unique output value (y).
• Cartesian Plane: A two-dimensional coordinate system used to graph functions.

Definition and Properties of Exponential Functions

• Definition: An exponential function is defined as f(x) = a^x, where 'a' is the base and 'x' is the exponent. The variable 'x' is in the exponent, which distinguishes it from polynomial functions like x^2.
• Conditions for the Base: The base 'a' must be a real number greater than zero (a > 0) and not equal to one (a ≠ 1).
• Domain: Exponential functions are defined for all real numbers (x ∈ ℝ).
• Applications: Exponential functions are used to model various phenomena, including compound interest in finance, bacterial growth, and the spread of diseases.
• Notation: The function can be written as f(x) = a^x or y = a^x.
• Injective Property: Exponential functions are injective, meaning that different values of x will always produce different values of f(x). If x1 ≠ x2, then f(x1) ≠ f(x2).
• Graph Behavior: The graph of an exponential function never crosses the x-axis; it approaches it asymptotically.
• Inverse Function: The inverse function of an exponential function is the logarithmic function.
• Property f(0) = 1: For any exponential function f(x) = a^x, f(0) = a^0 = 1.

Examples of Exponential Functions

• f(x) = 2^x (base is 2)
• f(x) = (1/3)^x (base is 1/3)
• f(x) = 5^x (base is 5)
• f(x) = (4/5)^x (base is 4/5)
• Functions of the type 3 + 2^x are considered exponential function types due to the presence of the exponential term.
• Non-Example: x^2 is a quadratic function, not an exponential function, because the variable 'x' is in the base, not the exponent.

Calculating Numerical Values

• Example 1: Given f(x) = 3^x, calculate f(4).
  • Substitute x = 4 into the function: f(4) = 3^4 = 3 * 3 * 3 * 3 = 81.
• Example 2: Given f(x) = 3^x, calculate f(-2).
  • Substitute x = -2 into the function: f(-2) = 3^(-2) = 1 / (3^2) = 1/9.
• Example 3: Given f(x) = 3^x and f(x) = 27, find x.
  • Set up the equation: 3^x = 27.
  • Express 27 as a power of 3: 27 = 3^3.
  • Equate the exponents: x = 3.
• Fractional Base Example 1: Given f(x) = (1/5)^x, calculate f(2).
  • Substitute x = 2 into the function: f(2) = (1/5)^2 = (1/5) * (1/5) = 1/25.
• Fractional Base Example 2: Given f(x) = (1/5)^x, calculate f(-3).
  • Substitute x = -3 into the function: f(-3) = (1/5)^(-3) = 5^3 = 125.
• Fractional Base Example 3: Given f(x) = (1/5)^x and f(x) = 625, find x.
  • Set up the equation: (1/5)^x = 625.
  • Express 625 as a power of 5: 625 = 5^4.
  • Rewrite the equation: (1/5)^x = 5^4.
  • Invert the base on the left side: (5^(-1))^x = 5^4 => 5^(-x) = 5^4
  • Equate the exponents: -x = 4 => x = -4.

Types of Exponential Functions: Increasing vs. Decreasing

• Increasing Function: If the base 'a' is greater than 1 (a > 1), the function is increasing. As x increases, f(x) increases. Example: f(x) = 2^x, f(x) = 1.4^x.
• Decreasing Function: If the base 'a' is between 0 and 1 (0 < a < 1), the function is decreasing. As x increases, f(x) decreases. Example: f(x) = (1/5)^x, f(x) = 0.5^x.
• Identifying Increasing/Decreasing: To determine if a function is increasing or decreasing, examine the base 'a'. If 'a' is a fraction, convert it to a decimal to easily see if it falls between 0 and 1.

Graphing Exponential Functions

• Creating a Table of Values: Choose several x-values (including negative, zero, and positive values) and calculate the corresponding y-values (f(x)).
• Plotting Points: Plot the ordered pairs (x, y) on the Cartesian plane.
• Connecting the Points: Draw a smooth curve through the plotted points. The graph will approach the x-axis but never cross it.
• Domain and Range:
  • The domain of an exponential function is all real numbers (x ∈ ℝ).
  • For f(x) = a^x, the range is all positive real numbers (y > 0).
• Example: Graphing f(x) = 2^x
  • Create a table:
    • x = -1, y = 1/2
    • x = 0, y = 1
    • x = 1, y = 2
    • x = 2, y = 4
    • x = 3, y = 8
  • Plot the points and connect them to form an increasing curve.
• Example: Graphing f(x) = (1/3)^x
  • Create a table:
    • x = -2, y = 9
    • x = -1, y = 3
    • x = 0, y = 1
    • x = 1, y = 1/3
    • x = 2, y = 1/9
  • Plot the points and connect them to form a decreasing curve.

Solving Exponential Equations

• Goal: To solve an exponential equation, rewrite the equation so that both sides have the same base.
• Method:
  1. Express both sides of the equation with the same base.
  2. Equate the exponents.
  3. Solve for the variable.
• Example: Solve 3^x = 27.
  1. Rewrite 27 as 3^3: 3^x = 3^3.
  2. Equate the exponents: x = 3.

Conclusion

Exponential functions are powerful tools for modeling various real-world phenomena. Understanding their properties, including the base, exponent, domain, range, and increasing/decreasing behavior, is crucial for working with these functions. Being able to calculate numerical values, graph exponential functions, and solve exponential equations provides a solid foundation for applying these concepts in different contexts. The key takeaway is that the base dictates whether the function increases or decreases, and the variable resides in the exponent, distinguishing it from other types of functions.