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Functions and Logarithms

This section reviews essential concepts of functions and logarithms, focusing on the Vertical Line Test (VLT) and Horizontal Line Test (HLT) to determine if relations are functions and if their inverses are also functions. One-to-One functions uniquely map inputs to outputs, making them reversible. The inverse function can be found by switching x and y, confirming through composition. Additionally, the relationship between exponential and logarithmic forms is explored, emphasizing logarithms' significance and properties, including the Change of Base formula and practical evaluation.

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Functions and Logarithms

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  1. Functions and Logarithms Section 1.5

  2. First, some basic review… What does the Vertical Line Test tell us? Whether or not the graph of a relation is a function… What does the Horizontal Line Test tell us? Whether or not a relation’s inverse is a function… A function that passes both the VLT and HLT is called: One-to-One

  3. One-to-One Functions

  4. Inverses • Since each output of a one-to-one function comes from just one input, a one-to-one function can be reversed to send outputs back to the inputs from which they came. • The function defined by reversing a one-to-one function f is the inverse of f. • Composing a function with its inverse in either order sends each output back to the input from which it came.

  5. Inverses

  6. Inverses Find and verify that First, solve for x: Interchange x and y: Verification: (in the given domains)

  7. Logarithmic Functions Switching between exponential and logarithmic form: The key element of this “linking” statement: Logarithms ARE Exponents!!!

  8. Logarithmic Functions • Logarithms with base e and base 10 are so important in applications that calculators have special keys for them. • They also have their own special notations and names.

  9. Logarithmic Functions How to “read” logarithmic expressions… “b to what power is x?” “Read” each properly, then evaluate: “2 to what power is 8?” “10 to what power is 1/100?” 5 “e to what power is e ?”

  10. Properties of Logarithms Change of Base (COB) Formula

  11. Guided Practice Find a formula for Switch x and y, solve for y:

  12. Guided Practice Solve the equation algebraically. Support graphically. Does the graph support our analytic answer?

  13. Guided Practice Solve the equation algebraically. Support graphically. This equation is quadratic in ...

  14. Guided Practice Solve the equation algebraically. Support graphically. Do both of these answers work? Graphical support?

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