Download presentation

1
**3.3 Perform Function Operations & Composition**

What is the difference between a power function and a polynomial equation? What operations can be performed on functions? What is a composition of two functions? How is a composition of functions evaluated?

2
**Operations on Functions: for any two functions f(x) & g(x)**

Addition h(x) = f(x) + g(x) Subtraction h(x) = f(x) – g(x) Multiplication h(x) = f(x)*g(x) OR f(x)g(x) Division h(x) = f(x)/g(x) OR f(x) ÷ g(x) Composition h(x) = f(g(x)) OR g(f(x)) ** Domain – all real x-values that “make sense” (i.e. can’t have a zero in the denominator, can’t take the even nth root of a negative number, etc.)

3
Power Functions

4
**Domain of (a) all real numbers Domain of (b) all real numbers**

Ex: Let f(x)=3x1/3 & g(x)=2x1/3. Find (a) the sum, (b) the difference, and (c) the domain for each. 3x1/3 + 2x1/3 = 5x1/3 3x1/3 – 2x1/3 = x1/3 Domain of (a) all real numbers Domain of (b) all real numbers

5
**Let f (x) = 4x1/2 and g(x) = –9x1/2. Find the following.**

f(x) + g(x) SOLUTION f (x) + g(x) = [4 + (–9)]x1/2 = –5x1/2 b. f(x) – g(x) SOLUTION f (x) – g(x) = 4x1/2 – (–9x1/2) = [4 – (–9)]x1/2 = 13x1/2 c. the domains of f + g and f – g The functions f and g each have the same domain: all nonnegative real numbers. So, the domains of f + g and f – g also consist of all nonnegative real numbers.

6
Types Domains All real numbers – if you can use positive numbers, negative numbers, or zero in the beginning functions and the result of combining functions. All non-negative numbers -- if you can use positive numbers and zero in the beginning functions and the result of combining functions. All positive numbers -- if you can use only positive numbers in the beginning function and the result of combining functions

7
**Ex: Let f(x)=4x1/3 & g(x)=x1/2**

Ex: Let f(x)=4x1/3 & g(x)=x1/2. Find (a) the product, (b) the quotient, and (c) the domain for each. 4x1/3 * x1/2 = 4x1/3+1/2 = 4x5/6 = 4x1/3-1/2 = 4x-1/6 = (c) Domain of (a) all reals ≥ 0, because you can’t take the 6th root of a negative number (Non-neg #’s). Domain of (b) all reals > 0, because you can’t take the 6th root of a negative number and you can’t have a denominator of zero (Positive #’s).

8
**Let f (x) = 6x and g(x) = x3/4. Find the following.**

SOLUTION f (x) g(x) = 6x x3/4 = 6x(1 – 3/4) = 6x1/4 SOLUTION the domain of The domain of f consists of all real numbers, and the domain of g consists of all nonnegative real numbers. Because g(0) = 0, the domain of is restricted to all positive real numbers.

9
**Try it… Let f (x) = –2x2/3 and g(x) = 7x2/3. Find the following.**

f (x) + g(x) SOLUTION f (x) + g(x) = –2x2/3 + 7x2/3 = (–2 + 7)x2/3 = 5x2/3 f (x) – g(x) SOLUTION f (x) – g(x) = –2x2/3 – 7x2/3 = [–2 + ( –7)]x2/3 = –9x2/3 The domains of f and g have the same domain: all non-negative real numbers. So , the domain of f + g and f – g also consist of all non-negative real numbers.

10
**Composition of Functions**

11
**Ex: Let f(x)=2x-1 & g(x)=x2-1**

Ex: Let f(x)=2x-1 & g(x)=x2-1. Find (a) f(g(x)), (b) g(f(x)), (c) f(f(x)), and (d) the domain of each. (a) 2(x2-1)-1 = (c) 2(2x-1)-1 = 2(2-1x) = (b) (2x-1)2-1 = 22x-2-1 = (d) Domain of (a) all reals except x=±1. Domain of (b) all reals except x=0. Domain of (c) all reals except x=0, because 2x-1 can’t have x=0.

12
**Let f(x) = 3x – 8 and g(x) = 2x2. Find the following.**

g(f(5)) SOLUTION To evaluate g(f(5)), you first must find f(5). f(5) = 3(5) – 8 = 7 Then g( f(3)) = g(7) = 2(7)2 = 2(49) = 98. So, the value of g(f(5)) is 98. ANSWER

13
**Let f(x) = 3x – 8 and g(x) = 2x2. Find the following.**

f(g(5)) SOLUTION To evaluate f(g(5)), you first must find g(5). g (5) = 2(5)2 = 2(25) = 50 Then f( g(5)) = f(50) = 3(50) – 8 = 150 – 8 = 142. So, the value of f( g(5)) is 142. ANSWER

14
**Let f(x) = 2x–1 and g(x) = 2x + 7. Find f(g(x)), g(f(x)), and f(f(x))**

Let f(x) = 2x–1 and g(x) = 2x + 7. Find f(g(x)), g(f(x)), and f(f(x)). Then state the domain of each composition. SOLUTION = 2 2x + 7 f(g(x)) = f(2x + 7) = 2(2x + 7)–1 g(f(x)) = f(2x–1) 4 x = + 7 = 2(2x–1) + 7 = 4x–1 + 7 f(f(x)) = f(2x–1) = 2(2x–1)–1 = x The domain of f(g(x )) consists of all real numbers except x = –3.5. The domain of g(f(x)) consists of all real numbers except x = 0.

15
**What is the difference between a power function and a polynomial equation?**

The power tells you what kind of equation—linear, quadratic, cubic… What operations can be performed on functions? Add, subtract, multiply, divide. What is a composition of two functions? A equation (function) is substituted in for the x in another equation (function). How is a composition of functions evaluated? Write the outside function and substitute the other function for x.

16
**Page 184, 3-27 every 3rd problem, 29-35 odd**

Assignment Page 184, 3-27 every 3rd problem, odd

Similar presentations

© 2020 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google