1.5 Functions and Logarithms AP Calculus AB/BC 1.5 Functions and Logarithms

A relation is a function if: for each x there is one and only one y. A relation is a one-to-one if also: for each y there is one and only one x. In other words, a function is one-to-one on domain D if: whenever

To be one-to-one, a function must pass the horizontal line test as well as the vertical line test. not one-to-one not a function (also not one-to-one)

Example 1 − Inverse functions: Given an x value, we can find a y value. Solve for x: Inverse functions are reflections about y = x. Switch x and y: (eff inverse of x)

Example 1 − Inverse functions:

Example 2 − Inverse functions: Finally, compose the two functions to verify they are inverses of each other. Now, solve for x in terms of y. Start by multiplying both sides by x + 3. Next, collect all the x’s together. Next, factor. Then, divide.

Example 3 Graph: for WINDOW a parametrically: Y= GRAPH

b Find the inverse function: Example 3 Graph: for b Find the inverse function: WINDOW Switch x & y: Change the graphing mode to function. Y= TEST 3: > GRAPH

Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function. Example: Two raised to what power is 16? The most commonly used bases for logs are 10: and e: is called the natural log function. is called the common log function.

In calculus we will use natural logs exclusively. We have to use natural logs: Common logs will not work. is called the natural log function. is called the common log function.

Even though we will be using natural logs in calculus, you may still need to find logs with other bases occasionally. Here are some useful examples using logs of different bases with the TI-84. LOG 1000) ENTER = 3 LN 32) ÷ LN 2) ENTER = 5 The second example illustrates change of base formula for logarithms. The next slide revies the logarithm properties.

Properties of Logarithms Since logs and exponentiation are inverse functions, they “un-do” each other. Product rule: Quotient rule: Power rule: Change of base formula:

Example 4 Solve for y: First, exponentiate both sides. Next, use the inverse property.

Example 5 Solve the equation algebraically. First, take the natural log of both sides. Next, use the power property to bring the power to the front. Finally, divide both sides by ln(1.045).

Example 6 Draw the graph and determine the domain and range of the function. First, rewrite the function using the change of base formula. Next, type the new equation into your calculator. Set the window at: Finally, graph the function.

Example 7 Draw the graph and determine the domain and range of the function. First, rewrite the function using the change of base formula. Next, type the new equation into your calculator. Set the window at: Finally, graph the function.

p* Example 8 $1000 is invested at 5.25 % interest compounded annually. How long will it take to reach $2500? We use logs when we have an unknown exponent. * Stop here, or continue to include regression functions on the TI-89. 17.9 years In real life you would have to wait 18 years. p*

Example 9 Indonesian Oil Production (million barrels per year): Use the natural logarithm regression equation to estimate oil production in 1982 and 2000. How do we know that a logarithmic equation is appropriate? In real life, we would need more points or past experience.

The calculator gives you an equation and constants: Indonesian Oil Production: 2nd { 60,70,90 2nd } 60 70 90 20.56 million 42.10 70.10 ENTER STO 2ND L1 The calculator should return: STAT CALC 9:LnReg ENTER The calculator gives you an equation and constants:

We can use the calculator to plot the new curve along with the original points: Y= Y1=RegEQ ENTER VARS 5:Statistics EQ 1:RegEQ 4 times ENTER Plot 1 WINDOW GRAPH

p What does this equation predict for oil production in 1982 and 2000? TRACE This lets us see values for the distinct points. This lets us trace along the line. 82 ENTER Enters an x-value of 82. Moves to the line. In 1982, production was 59 million barrels. 100 ENTER Enters an x-value of 100. p In 2000, production was 84 million barrels.