Exponential Functions and Their Properties

The video tutorial introduces the concept of exponential growth and decay, explaining how certain phenomena like radioactive decay and compound interest follow exponential patterns. It defines exponential functions as those with a constant base raised to a variable exponent, emphasizing the importance of the base being positive and not equal to one. Through a game show format, the video helps viewers identify exponential functions, distinguishing them from polynomials and other non-exponential forms. Finally, it provides practice in evaluating exponential functions, including handling negative and fractional exponents.

Which of the following is an example of exponential growth?

• A car depreciating in value

• A bank account earning compound interest

• A person aging

• A tree growing in height

What is the key characteristic of an exponential function?

• The variable is in the base

• The variable is in the exponent

• The base is always 1

• The exponent is always a constant

Which of the following is NOT an exponential function?

• y = 2^x

• y = 3^(x+1)

• y = x^2

• y = 5^(-x)

Why can't the base of an exponential function be negative?

• It would always result in negative numbers

• It would result in complex numbers for some exponents

• It would always result in zero

• It would always result in positive numbers

What is the value of f(2) for the function f(x) = 3^x?

• 12

• 3

• 6

• 9

How do you handle a negative exponent in an exponential function?

• Subtract the base

• Add the base

• Take the reciprocal of the base

• Multiply by the base

What is the value of f(2) for the function f(x) = 4^(-x)?

• 1/32

• 1/8

• 1/4

• 1/16

How do you evaluate an exponential function with a fractional exponent?

• Raise the base to the power of the numerator and then take the root indicated by the denominator

• Take the root indicated by the denominator and then raise to the power of the numerator

• Square the base

• Take the reciprocal of the base

What is the value of f(3/2) for the function f(x) = 25^x?

• 25

• 50

• 125

• 150

What is the value of f(-3) for the function f(x) = (1/2)^x?

• 1/4

• 1/8

• 4

• 8