6.3 Power Functions & 6.3 Power Functions & Function OperationsFunction Operations

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?

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

1.1. AdditionAddition h(x) = f(x) + g(x)

2.2. SubtractionSubtraction h(x) = f(x) – g(x)

3.3. MultiplicationMultiplication h(x) = f(x)*g(x) OR f(x)g(x)

4.4. DivisionDivision h(x) = f(x)/g(x) OR f(x) ÷ g(x)

5.5. CompositionComposition h(x) = f(g(x)) OR g(f(x))

** DomainDomain – 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.)

Let f(x)=3xLet f(x)=3x1/31/3 & g(x)=2x & g(x)=2x1/31/3. . Find (a) f(x) + g(x) Find (a) f(x) + g(x) (b) f(x) – g(x)(b) f(x) – g(x)(c) the domain for each.(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 numbers

Domain of (b) all real numbers

Let f(x)=4x1/3 & g(x)=x1/2. Find . Find (a) the product, (b) the quotient, and (c) (a) the product, (b) the quotient, and (c) the domain for each.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

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

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

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

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 Domain of (a) all reals except x=x=±1.±1.

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

Domain of (c) all reals except x=0, Domain of (c) all reals except x=0, because 2xbecause 2x-1-1 can’t have x=0. can’t have x=0.

• 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.