Answers to Questions on Exponential Functions


  1. Question: Which of the following are exponential functions?

    1. f(x) = 3e-2 x

    2. g(x) = 2x/2

    3. h(x) = x3/2

    4. g(x) = 15 / 7x

    Answer: Of these functions, only h(x) is not an exponential function. Remember that the independent variable must appear in the exponent for the function to be exponential.

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  2. Question: What is the domain of an exponential function f(x) = kbx? What is the range? Describe the shape of the graph for b > 1, and for b < 1. What happens to f(x) in each case when x becomes very large (increases without bound) and as x becomes very small (decreases without bound)? Are there any horizontal asymtotes?

    Answer: The domain of an exponential function of this form is all real numbers. The range (co-domain) is all positive real numbers.

    For b > 1, f(x) is increasing -- its graph rises to the right. As x increases without bound, so does f(x), but as x decreases without bound, f(x) approaches zero. The line y = 0 (the x-axis) is a horizontal asymptote.

    For b < 1, f(x) is decreasing -- its graph falls to the right. As x increases without bound, f(x) approaches zero, so the x-axis is a horizontal asymptote. On the other hand, as x decreases without bound, f(x) grows infinitely large. The left end goes up and the right end slides along the x-axis.

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  3. Approximate values are given to 4 decimal places.

    1. Question: 2x = 32, What is x?
      Answer: x = 5
      Explain

    2. Question: 5x = 1/125, What is x?
      Answer: x = -3 Explain

    3. Question: (1/3)2x = 243, What is x?
      Answer: x = -5/2 Explain

    4. Question: 45 = 53x, What is x?
      Answer: x = 0.7884 Explain

    5. Question: 500 = 1000e-.75 x, What is x?
      Answer: x = 0.9242 Explain

    6. Question: 56 = 14(1 + e.195 x), What is x?
      Answer: x = 5.6339 Explain

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  4. Question: The population of bacteria in a culture is growing exponentially. At 12:00 there were 80 bacteria present and by 4:00 PM there were 500 bacteria. Find an exponential function f(t) = ke at that models this growth, and use it to predict the size of the population at 8:00 PM.

    Answer: The exponential function is f(t) = 80 e.4581 t. There will be 3,125 bacteria at 8:00 PM. Explain

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  5. Question: The last nuclear test explosion was carried out by the French on an island in the south Pacific in 1996. Immediately after the explosion, the level of strontium-90 on the island was 100 times the level considered to be "safe" for human habitation. If the half-life of Strontium-90 is 28 years, how long will it take for the island to once again be habitable?

    Answer: The island will be safe in about 186 years, that is, in the year 2182.Explain

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