Section 9.3 We have previously worked with exponential expressions, where the exponent was a rational number The expression b x can actually be defined for all real numbers, x, including irrational numbers. However, the proof of this would have to wait until a higher level math course.

A function of the form f(x) = b x is called an exponential function if b > 0, b is not 1, and x is a real number.

We can graph exponential functions of the form f(x) = 3 x, g(x) = 5 x or h(x) = (½) x by substituting in values for x, and finding the corresponding function values to get ordered pairs. We would find all graphs satisfy the following properties: 1-to-1 function y-intercept (0, 1) no x-intercept domain is (- ,  ) range is (0,  )

We would find a pattern in the graphs of all the exponential functions of the type b x, where b > 1. x y

We would find a pattern in the graphs of all the exponential functions of the type b x, where 0 < b < 1. x y

We would find a pattern in the graphs of all the exponential functions of the type b x-h, where b > 1. x y (h, 1) The graph has the same shape as the graph for b x, except it is shifted to the right h units.

We would find a pattern in the graphs of all the exponential functions of the type b x+h, where b > 1. x y (-h, 1) The graph has the same shape as the graph for b x, except it is shifted to the left h units.

Since an exponential function is a 1-to-1 function, if b > 0, and b  1, then b x = b y is equivalent to x = y. Example Solve 6 x = 36 6 x = 6 2 x = 2

Example Solve 9 2x+1 = x+1 = 9 2 2x + 1 = 2 2x = 1 x = ½

Example Solve 3 -3 = 3 2x -3 = 2x

Example Solve 4 3x-6 = 32 2x (2 2 ) 3x-6 = (2 5 ) 2x (2 2 ) 3x-6 = 2 10x 2 6x-12 = 2 10x 6x-12 = 10x -12 = 4x x = -3

Many applications use exponential functions of various types. Compound interest formulas are exponential functions used to determine the amount of money accumulated or borrowed. Exponential functions with negative exponents can be used to describe situations of decay, while those with positive exponents can be used to describe situations of growth.

Example Find the total amount invested in a savings account if $5000 was invested and earned 6% compounded monthly for 18 years. Round your answer to two decimal places. The formula that is used for calculating compound interest is where P is the initial principal invested, r is the interest rate, n is the number of times interest is compounded each year, t is the time of the investment (in years) and A is the amount of money in the account.

Example (cont.) $

Example An accidental spill of 100 grams of radioactive material in a local stream has led to the presence of radioactive debris decaying at a rate of 5% each day. Find how much debris still remains after 30 days. The formula that would be used for this problem is where A is the amount of radioactive material to start, r is the rate of decay, t is the number of days and y is the amount of radioactive material after the time period.

Example (cont.) (exact answer)