7.3 Power Functions and Function Operations Goals Perform operations with functions, including power functions. Use power functions and function operations to solver real-life problems.

Evaluate each function for when x = -2 f(x) = 2x + 5 f(x) = 2(-2) + 5 f(x) = -4 + 5 f(x) = 1 g(x) = x2 – 1 g(x) = (-2)2 – 1 g(x) = 4 -1 g(x) = 3

Domain The numbers that CAN be plugged in for x in a function Have to look at beginning function through the ending function for domain!!! Try to find the values that CANNOT be plugged in to determine domain 1. Can’t divide by zero (undefined) 2. Can’t take even root of a negative number If there are no restrictions on x, then the domain is “all reals”

Operations with Functions The four basic operations can be applied to any function. Let’s perform the basic operations with this example: f(x) = 2x + 5 g(x) = x2 - 1

Addition of Functions EX: f(x) = 2x + 5 g(x) = x2 - 1 f(x) + g(x) = (2x + 5) + (x2 - 1) = 2x + 5 + x2 - 1 = x2 + 2x + 4 Domain = “all reals” Is f(x) + g(x) = g(x) + f(x)?

Subtraction of Functions EX: f(x) = 2x + 5 g(x) = x2 - 1 f(x) – g(x) = (2x + 5) – (x2 - 1) = 2x + 5 – x2 + 1 = -x2 + 2x + 6 Domain = “all reals” Is f(x) – g(x) = g(x) – f(x)?

Multiplication of Functions EX: f(x) = 2x + 5 g(x) = x2 - 1 f(x) · g(x) = (2x + 5)(x2 - 1) = 2x3 - 2x + 5x2 - 5 = 2x3 +5x2 - 2x - 5 Domain = “all reals” Is f(x) · g(x) = g(x) · f(x)? ·

Division of Functions EX: f(x) = 2x + 5 g(x) = x2 - 1 f(x) / g(x) = (2x + 5) / (x2 - 1) Domain = Is f(x) / g(x) = g(x) / f(x)? ·

Composition Plug one function into another Put one function in place of x

Composition of Functions: Example 1 f(x) = 2x + 3 and g(x) = x2 + 5 Find: f(g(x)) f(x2 + 5) = 2(x2 + 5) + 3 = 2x2 + 10 + 3 = 2x2 + 13 Domain = “all reals”

Composition of Functions: Example 2 f(x) = 2x + 3 and g(x) = x2 + 5 Find: g(f(x)) g(2x + 3) = (2x + 3)2 + 5 = (2x + 3)(2x + 3) + 5 = 4x2 + 6x + 6x + 9 + 5 = 4x2 + 12x +14 Domain = “all reals”

Composition of Functions: Example 3 f(x) = 2x + 3 and g(x) = x2 + 5 Find: f(f(x)) f(2x + 3) = 2(2x + 3) + 3 = 4x + 6 + 3 = 4x + 9 Domain = “all reals”

One More Example f(x) = 3x and g(x) = x1/4 f(x) / g(x) Domain = “positive real #s” ·

Review and some more! Domain: look at beginning through ending Can’t divide by zero Can’t take even root of a negative number Difference between “nonnegative” and “positive” real numbers Nonnegative Real Numbers Includes “0” Positive Real Numbers Does NOT include “0”

EX: f(x) = 2x + 5 g(x) = x2 - 1 f(x) + g(x) = (2x + 5) + (x2 - 1) = 2x + 5 + x2 - 1 = x2 + 2x + 4 Domain = “all reals” EVALUATE for x = -4 = (-4)2 + 2(-4) + 4 = 16 - 8 + 4 = 12

Homework P.418 #28-31 all (answer questions & evaluate each for x=3)

7.3 Power Functions and Function Operations Day 2

Go over: - 28 - 41 - 30 - 44 - 31 - 47

Interval Notation Brackets mean > or < [ ] Parenthesis mean > or < ( ) Interval Notation

Domain for 3 more….. Positive v. Nonnegative

Homework 7.3 WS Practice B #2 - 16 even #17 - 22 all (answer question & evaluate each problem for x=2) Write out domain for ALL in “Interval Notation”