4-1 Exponential Functions

1. What is the definition of an exponential function? Page 412

An exponential function f with base b is defined by f(x) = bx or y = bx, where b is a positive constant other than 1 (b > 0 and b is not equal to 1) and x is any real number.

Example: g(x) = 10^x

2. What is the inverse of an exponential function? Page

No horizontal line can be drawn that intersects the graph of an exponential function at more than one point. This means that the exponential function is one-to-one and has an inverse. Example: fx = b(x) Steps for solving problem: Replace: fx with y: y = b(x) Interchange x and y: x=b(y) Solve for y

3. What are the characteristics of an exponential function? Page 415 * The domain of f(x) = b^x consists of all real numbers. The range of F(x) = b^x consists of all positive real numbers (0, to infinity). * The graphs of all exponential functions of the form f(x) = b^x pass Through the point (0, 1) because f(0) = b^0 =1 (b not equal to 0) the y intercept is 1. There is no x intercept. * If b > 1, f(x) – b^x has a graph that goes up to the right and is an increasing function. The greater the value of b, the steeper the increase. * If 0 < b < 1, f (x) = b^x has a graph that goes down to the right and is a decreasing function. The smaller the value of b, the steeper the decrease. * F(x) = b^x is one-to-one and has an inverse that is a function. * The graph of f(x) = b^x approaches, but does not touch, the x-axis. The x-axis, or y = 0, is horizontal asymptote.

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