7.3 – Power Functions & Function Operations

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

1. Addition h(x) = f(x) + g(x)

2. Subtraction h(x) = f(x) – g(x)

3. Multiplication h(x) = f(x) · g(x) OR f(x)g(x)

4. Division h(x) = f(x)/g(x) OR f(x) ÷ g(x)

Something New…

• 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.)

Ex: Let f(x) = 3x1/3 & g(x) = 2x1/3. Find (A) f(x) + g(x), (B) f(x) – g(x), and (C) the domain for each.

A. 3x1/3 + 2x1/3 = 5x1/3

B.3x1/3 – 2x1/3 = x1/3

C.Domain of (a) all real numbersDomain of (b) all real numbers

Ex: Let f(x) = x² + 4x – 3 and g(x) = x² - 1. Find (A) f(x) + g(x), (B) f(x) – g(x), (C) g(x) – f(x), and (d) the domain for

each.

A.2x² + 4x – 4 Domain: All real #’s

B.4x – 2 Domain: All real #’s

C.-4x + 2 Domain: All real #’s

Note: The domain of the resulting function is determined by the functions being used, not just the resulting function itself.

Example: xxf )( xxg )(

1)(

)()(

x

x

xg

xfx

g

f

0)()())(( xxxgxfxgf

Domain of (f - g) is all non-negative real numbers.

Domain of (f /g) is all positive real numbers.

Ex: Let f(x) = 4x1/3 & g(x) = x1/2. Find (A) the product, (B) the quotient, and (C) the domain for each.

A. 4x1/3 · x1/2 = 4x1/3+1/2 = 4x5/6

B.

= 4x1/3-1/2 = 4x-1/6 =

2

1

3

1

4

x

x

6

1

4

x

C. Domain of (A) all reals ≥ 0, because you can’t take the 6th root of a negative number.

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.

564 x

6

4

x

Ex: Let f(x) = 3x and g(x) = x1/4. Find (A) f(x) · g(x), (B) f(x) ÷ g(x),

and (C) the domain for each.

A.3x5/4 Domain: All positive real #’s

B.3x3/4 Domain: All positive real #’s

Homework

Power Functions

!

pgs. 418-419 #32-35, 40-46 even

Quiz 7.1-7.3 on Thursday, February 14th

7.3 – Power Functions & Function Operations

(Day 2)

Objectives:• Today we will…

– Clarify and expand our understanding of finding domains.

– Be introduced to the composition of functions.

– Explain our reasoning behind why we solve problems a certain way.

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

1. Addition h(x) = f(x) + g(x)2. Subtraction h(x) = f(x) – g(x)3. Multiplication h(x) = f(x) · g(x) OR f(x)g(x)4. Division h(x) = f(x)/g(x) OR f(x) ÷ g(x)

5. Composition h(x) = f(g(x)) or h(x) = 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.)

REVIEW: The domain of the resulting function is determined by the functions being used, not just the resulting function itself.

Example: xxf )( xxg )(

1)(

)()(

x

x

xg

xfx

g

f

0)()())(( xxxgxfxgf

Domain of (f - g) is all non-negative real numbers.

Domain of (f /g) is all positive real numbers.

Composition• f(g(x)) means you take the function

g and plug it in for the x-values in the function f, then simplify.

• g(f(x)) means you take the function f and plug it in for the x-values in the function g, then simplify.

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 =1

22 x

B. (2x-1)2 – 1

= 22x-2 – 1

= 142

x

C. 2(2x-1)-1

= 2(2-1x)

=2

2x x

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.

Ex: Let f(x) = 3x-1 & g(x) = 2x – 1. Find (A) f(g(x)), (B) g(f(x)), (C) f(f(x)), and (d) the domain of each.

A. 3(2x – 1)-1 =3

2 1x

B. 2(3x-1) – 1

= 6x-1 – 1

= 6 1x

C. 3(3x-1)-1

= 3(3-1x)

= 3

3

x x

D. Domain of (A) all reals except x = -1/2.

Domain of (B) all reals except x = 0.

Domain of (C) all reals except x = 0 because 3x-1 can’t have x = 0.

Homeworkpgs. 418-419 #36-39, 48-51

Quiz 7.1-7.3 on Thursday, February 14th

Power Functions!