5 x2 + 4x + 7 25 + 3 = 28 1.6 Warm-Up Evaluate the compositions if: f(x) = x + 2 g(x) = 3 h(x) = x2 + 3 1. f(g(x)) 2. h(f(x)) 3. h(f(g(x))) f(3) h(x + 2) h(f(3)) f(x) = x + 2 h(x) = x2 + 3 h(3 + 2) 3 + 2 (x + 2)2 + 3 h(5) 52 + 3 5 x2 + 4x + 4 + 3 x2 + 4x + 7 25 + 3 = 28

1.6a Inverse Functions Objective – Students will define one-to-one functions and use the horizontal line test. Students will be able to solve problems involving inverse functions. To be able to evaluate logarithmic functions.

1.6a Inverse Functions Definition: A function f is called one-to-one if it never takes on the same value twice; that is... whenever Horizontal Line Test: A function is one-to-one if and only if no horizontal line intersects its graph more than once

Solving for the Inverse 1.6a Inverse Functions Solving for the Inverse STEP 1 Switch the “y” and the “x” values. STEP 2 Solve for “y”.

Answer: y -1 = -5x + 2 Example 1: Find the inverse of 10y +2x = 4 10x + 2y = 4 2y = -10x + 4 2 2 y = -5x + 2 Answer: y -1 = -5x + 2

Answer: y -1 = (-1/3)x + 2 Example 2: Find the inverse of y = -3x + 6 x = -3y + 6 x = -3y + 6 –6 –6 x – 6 = -3y –3 –3 y = (-1/3)x + 2 Answer: y -1 = (-1/3)x + 2

y = x 5 x = y 5 Example 3: Find the inverse of the function: f(x) = x5

Find the Inverse Equation: 1.6 Classwork Find the Inverse Equation: Y = 2x + 3 y = x2 + 1 y = x3 – 1 f(x) = 1 x3 2x + 1 x + 3

1.6 Homework Pg. 74-75: (5-8, 17, 23-28) all

Logarithmic Functions https://www.youtube.com/watch?v=VSgB1IWr6O4 The Richter Scale Magnitude +1 0 1 2 3 4 5 6 7 8 9 E E(30) E(30)2 E(30)3 E(30)4 E(30)5 E(30)6 E(30)7 E(30)8 E(30)9 energy released: x 30

Logarithmic Functions DEFINITION OF LOGARITHM WITH BASE b logby = x if and only if bx = y The expression logby is read as “log base b of y.”

Example 4 Rewrite the logarithmic equation in exponential form. log3 9 = 2 b) log8 1 = 0 c) log5 (1/25) = -2 32 = 9 80 = 1 5(-2) = 1/25 Example 5 Evaluate the expression. log4 64 b) log3 27 c) log6 ( 1/36 ) 4 x = 64 3 x = 27 6 x = 1/36 x = -2 x = 3 x = 3

Properties of Logarithms Objective – To be able to use properties of logarithms State Standard – 11.0 Students will understand and use simple laws of logarithms. Product Property logb uv = logb u + logb v Quotient Property logb (u/v) = logb u - logb v Power Property logb un = n logb u Properties of Logarithms

Example 2 Expand log5 2x 6 log5 2 + log5 x 6 log5 2 + 6 log5 x

Example 3 Condense: 2 log3 7 – 5 log3 x = log3 7 2 – log3 x 5

1.6b Assignment Pg. 76 35, 36, 39, and 40