Exponential Functions Overview

Exponential Functions

• Exponential functions are defined by the form f(x) = b^x, where 'b' is a positive constant other than 1, and 'x' is any real number.
• The domain of an exponential function f(x) = b^x is all real numbers (ℝ).
• The range of an exponential function f(x) = b^x is (0, ∞).
• The function f(x) = e^x is called the natural exponential function.
• The irrational number e is approximately 2.72 and is considered the natural base.
• Graphs of exponential functions with b > 1 are increasing, going upwards to the right. The larger the value of 'b', the steeper the graph increases.
• Graphs of exponential functions with 0 < b < 1 are decreasing, going downwards to the right. The smaller the value of 'b', the steeper the graph decreases.
• Exponential functions are one-to-one and have inverse functions.
• The x-axis (y = 0) is a horizontal asymptote for all exponential functions.

Examples and Evaluation

• Examples of exponential functions include f(x) = 2^x, g(x) = 10^x, h(x) = π^x, and others.
• Evaluating an exponential function involves substituting a given value of 'x' into the function and calculating the corresponding 'f(x)' value.
• When evaluating, round the answer to three decimal places if necessary.

Graphing Exponential Functions

• The domain of an exponential function is all real numbers.
• The range is always positive values.
• The horizontal asymptote is always y = 0.
• The graph passes through (0, 1).
• The graph will either increase or decrease depending on the base.

Transformations of Exponential Functions

• Vertical shifts: Add or subtract a constant to the function's output (y-value).
• Horizontal shifts: Add or subtract a constant to the function's input (x-value).
• Reflections across the x-axis: Multiply the output by -1.
• Reflections across the y-axis: Multiply the input by -1.
• Vertical stretching/shrinking: Multiply the output by a constant 'c'. |c|>1 is a vertical stretch; |c|<1 is a vertical shrink
• Transformations to exponential functions change the graph and its asymptote. The asymptote will also shift based on vertical transformations

Characteristics of Exponential Functions

• The domain of exponential functions includes all real numbers.
• The range of exponential functions is all positive real numbers.
• The graph of every exponential function passes through the point (0, 1).
• Exponential functions are strictly increasing if the base is greater than 1.
• Exponential functions are strictly decreasing if the base is between 0 and 1.
• The x-axis is a horizontal asymptote for all exponential functions.

This quiz covers the key concepts of exponential functions, including their definition, domain, and range. You'll learn about the natural exponential function and explore the characteristics of graphs based on different values of 'b'. Test your understanding of these essential mathematical concepts.