6.1 Operations on Functions

E XAMPLE 1 Given f ( x ) = 3 x x and g ( x ) = 2 x 2 – x – 1, find ( f – g )( x ).

E XAMPLE 2 Given f ( x ) = 2 x x + 2 and g ( x ) = 3 x x – 4, find ( f + g )( x ).

E XAMPLE 3 Given f ( x ) = 3 x 2 – 2 x + 1 and g ( x ) = x – 4, find ( f ● g )( x ).

E XAMPLE 4 Given f ( x ) = 2 x x – 1 and g ( x ) = x + 2, find.

E XAMPLE 5 Given f ( x ) = x 2 and g ( x ) = Find the following. List any restrictions to domain or range. 1. ( f + g )( x )2. ( f – g )( x ) 3. ( f + g )( x )4.

E XAMPLE 6 If f(x) = 2x and g(x) = x 2 – 3x + 2 and h(x) = -3x – 4 then find each value. a.f[g(3)] b.g[h(-2)] c.h[f(-4)]

6.1 Part 2 – Composite Functions

C OMPOSITE F UNCTIONS Another way to combine functions is a composition of functions. In a composition of functions, the results of one function are used to evaluate a second function.

E XAMPLE 1 Find [ f ○ g ]( x ) for f ( x ) = 3 x 2 – x + 4 and g ( x ) = 2 x – 1. State the domain and range for each combined function.

E XAMPLE 2 Find [ g ○ f ]( x ) for f ( x ) = 3 x 2 – x + 4 and g ( x ) = 2 x – 1. State the domain and range for each combined function.

E XAMPLE 3 Find [ f ○ g ]( x ) and [ g ○ f ]( x ) for f ( x ) = x x + 3 and g ( x ) = x + 5. State the domain and range for each combined function.

E XAMPLE 4 If f (x) = ax m and g (x) = bx n, perform the operation stated. A) f (x)  g (x) B) f ( g (x)) C) g ( f (x))

E XAMPLE 5 Hector has $100 deducted from every paycheck for retirement. He can have this deduction taken before state taxes are applied, which reduces his taxable income. His state income tax is 4%. If Hector earns $1500 every pay period, find the difference in his net income if he has the retirement deduction taken before or after state taxes.