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Exponential Functions quiz #1

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  • What are the necessary conditions for a function to be classified as an exponential function, and how do you identify the base and exponent in such a function?
    A function is an exponential function if it has a constant, positive base (not equal to 1) and a variable exponent. The base is the fixed number being raised to a power, and the exponent is the variable. For example, in f(x) = 2^x, the base is 2 and the exponent is x.
  • Explain the significance of the number e in exponential functions and describe one real-world application where it is commonly used.
    The number e (approximately 2.71828) is a mathematical constant used as the base in exponential functions, especially in situations involving continuous growth or decay. One common real-world application of e is in calculating continuously compounded interest in finance.
  • What are the three necessary conditions for the base of an exponential function?
    The base must be constant, positive, and not equal to 1.
  • How do you identify the base and exponent in the function f(x) = 2^x?
    The base is 2, and the exponent is x.
  • What happens to the value of an exponential function when the exponent is negative?
    The function's value becomes a fraction, specifically 1 divided by the base raised to the positive exponent.
  • What is the horizontal asymptote of the graph of any basic exponential function f(x) = b^x?
    The horizontal asymptote is y = 0.
  • How does the value of the base b affect the direction and steepness of the graph of f(x) = b^x?
    If b > 1, the graph increases and gets steeper for larger b; if 0 < b < 1, the graph decreases and gets steeper for smaller b.
  • What is the domain and range of a basic exponential function f(x) = b^x?
    The domain is all real numbers, and the range is (0, ∞).
  • What is the number e, and how is it used in exponential functions?
    The number e is approximately 2.71828 and is used as a base in exponential functions, especially for modeling continuous growth or decay.
  • Give one real-world application where the exponential function with base e is commonly used.
    One common application is calculating continuously compounded interest in finance.