Evaluate the compositions if: f(x) = x + 2g(x) = 3h(x) = x 2 + 3 1. f(g(x))2. h(f(x))3. h(f(g(x))) f(3) 3 + 2 5 h(x + 2) (x + 2) 2 + 3 x 2 + 4x + 4 + 3.

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Presentation transcript:

Evaluate the compositions if: f(x) = x + 2g(x) = 3h(x) = x f(g(x))2. h(f(x))3. h(f(g(x))) f(3) h(x + 2) (x + 2) x 2 + 4x x 2 + 4x + 7 h(f(3)) h(3 + 2) h(5) = 28 f(x) = x + 2h(x) = x 2 + 3

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.

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

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

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

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

f -1 (x)= 5 x Example 3: Find the inverse of the function: f(x) = x 5 x = y 5 5 x = y y = x 5

Find the Inverse Equation: 1)Y = 2x + 3 2) y = x ) y = x 3 – 1 4) f(x) = 5) f(x) = 1 x 3 2x + 1 x + 3

Pg : (5-7, 17, 23-28) all

The Richter Scale Magnitude 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 log b y = x if and only if b x = y The expression log b y is read as “log base b of y.”

Example 4 Rewrite the logarithmic equation in exponential form. a)log 3 9 = 2 b) log 8 1 = 0 c) log 5 ( 1 / 25 ) = = = 15 (-2) = 1 / 25 Example 5 Evaluate the expression. a)log 4 64 b) log 3 27 c) log 6 ( 1 / 36 ) 4 x = 64 x = 3 3 x = 276 x = 1 / 36 x = 3 x = -2

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

log log 5 x 6 Example 2 Expand log 5 2x 6 log log 5 x

Example 3 Condense: 2 log 3 7 – 5 log 3 x = log – log 3 x 5 = log x 5 = log 3 49 x 5

Pg , 36, 39, and 40