2.10 Inverses of Exponential Functions

Overview of Inverses

  • The concept of inverses is fundamental in mathematics.

  • An inverse operation essentially reverses the effect of the original operation.

  • Example: The inverse of an exponential function is a logarithmic function.

Exponential Functions

  • Key features of exponential functions:

    • Example: An exponential function with base 2.

    • **Points: **

      • (-1.5, 1/12)

      • (0, 1)

      • (1, 2)

      • (2, 4)

      • (3, 8)

Logarithmic Functions

  • Logarithmic functions are the inverses of exponential functions.

  • For base 2 logarithm, notable points include:

    • (1, 0)

    • (2, 1)

    • (4, 2)

    • (8, 3)

    • This suggests a corresponding relationship between x and y values of logarithm and exponential functions.

Graphical Relationship

  • Graphs of exponential and logarithmic functions are reflections over the line y = x.

  • Asymptotes for:

    • Exponential functions occur on the x-axis.

    • Logarithmic functions occur on the y-axis.

Domains and Ranges

  • The domain of an exponential function correlates with the range of its logarithmic inverse, and vice versa.

    • Example Exponential:

      • Domain: -1, 0, 1, 2, 3

      • Corresponding Logarithmic Range: -1, 0, 1, 2

    • Exponential Range: 0.5, 1, 2, 4, 8 corresponds to Logarithmic Domain: -1, 0, 1, 2, 3

Properties of Inverses

  • Inverses undo each other:

    • Logs can undo exponents.

    • If f is an exponential function and g is its logarithmic inverse:

      • f(g(x)) = x

      • g(f(x)) = x

  • Composition of functions reinforces this relationship.

Identifying Exponential vs. Logarithmic Functions

  • Exponential functions have x-values that increase additively while y-values rise multiplicatively.

  • Logarithmic functions have x-values that increase multiplicatively while y-values rise additively.

  • Use this property to identify function types and find their inverses.

Finding Inverses

  • Inverse operation involves:

    • Swapping x and y values in the function.

    • Solving for the new y to express it in terms of x.

  • Example finding inverses:

    • For y = 3 log base 5 (x):

      • Swap y to x and solve for y.

      • Result: y = 5^(x/3).

    • For y = 10^(x/6):

      • Result: y = 2^(x/10).

Verifying Inverses

  • Test if functions are inverses by checking if f(g(x)) = x and g(f(x)) = x.

  • Example operational tests:

    • Plugging one function into another confirms their inverse relationships.

Practice Exercises

  • Students encouraged to attempt problems and discuss findings with peers or teachers for clarification.

Conclusion

  • Understanding inverses is crucial in mastering both exponential and logarithmic functions.

  • Practice is key! Ask questions if you face challenges.