Pre-­‐Calculus Mathematics 12 -­‐ 1.2 -­‐ Arithmetic Combinations of Functions Goal: Perform operations with functions both with and without the graph of the function Functions are number generators. When you put a value for the domain in the function, you will get the resulting value in the range. f ( x ) = x2 + 5 And just like numbers, functions can be added, subtracted, multiplied and divided. Use the following functions, f ( x ) = 2 x –1 & g ( x ) = x 2 – 4 to determine: a) Sum: ( f + g )( x ) = f ( x) + g ( x) b) Difference: ( f − g )( x ) = f ( x) − g ( x) Product: ( fg )( x ) = f ( x) ⋅ g ( x) Quotient: ⎛ f ⎞
f ( x)
⎜ ⎟ ( x ) =
g ( x)
⎝ g ⎠
Note: The domain of the new function must include the restrictions of the new functions as well as the restriction(s) of the original function(s) Example 1: Given functions below, determine each of the following. 1
2
f ( x) = 2 ,
g ( x) =
, h( x) = 2 x 2 − 5 x − 3 i ( x) = x 2 − 9,
x
x+2
a)
( gj )( −3)
j ( x) = x 2 ,
⎛ f ⎞
⎟ ( −4 )
⎝ g ⎠
b) ⎜
k ( x) = x
Pre-­‐Calculus Mathematics 12 -­‐ 1.2 -­‐ Arithmetic Combinations of Functions Example 2: Given functions below, determine each new combined function and its domain. 1
2
f ( x) = 2 ,
g ( x) =
, h( x) = 2 x 2 − 5 x − 3 i ( x) = x 2 − 9,
j ( x) = x 2 , k ( x) = x
x
x+2
a)
⎛ gk ⎞
b) ⎜ ⎟ ( x )
⎝ i ⎠
( g − f )( x )
Example 3: Use the graph of f ( x ) and g ( x ) to graph the following function. f ( x ) g ( x ) ( f − g )( x )