1.5 Functions and Logarithms. One-to-One Functions Inverses Finding Inverses Logarithmic Functions Properties of Logarithms Applications …and why Logarithmic.

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Presentation transcript:

1.5 Functions and Logarithms

One-to-One Functions Inverses Finding Inverses Logarithmic Functions Properties of Logarithms Applications …and why Logarithmic functions are used in many applications including finding time in investment problems. What you’ll learn about…

One-to-One Functions A function is a rule that assigns a single value in its range to each point in its domain. Some functions assign the same output to more than one input. Other functions never output a given value more than once. If each output value of a function is associated with exactly one input value, the function is one-to-one.

One-to-One Functions

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.

Inverses

Identity Function The result of composing a function and its inverse in either order is the identity function.

Example Inverses

Writing f -1 as a Function of x.

Finding Inverses

Example Finding Inverses [-10,10] by [-15, 8]

Base a Logarithmic Function

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.

Inverse Properties for a x and log a x

Properties of Logarithms

Example Properties of Logarithms

Change of Base Formula

Example Population Growth