1. Identify whether or not each relation is a function. a. {(2,3), (4,5), (6,6)} b. {(4,5), (4,6), (5,5), (5,6)} c. {(6,7), (6,8), (7,7), (7,8)} 2. Evaluate each function at the indicated values of the independent variable and simplify the result. a. f(x) = 9 - 6x; 1. f(-1) 2. f(1) 3. f(-3 + x) b. g(x) = x^2 - 4x; 1. g(2) 2. g(2 - x) 3. Find (f + g)(x), (f - g)(x), (f · g)(x) and (f/g)(x). f(x) = 3x + 4; g(x) = 2x - 1
Answer (1)
Determine if each relation is a function:
A relation is a function if every input (or first element in each pair) is associated with exactly one output (or second element).
(a) {( 2 , 3 ) , ( 4 , 5 ) , ( 6 , 6 )}
Each input (2, 4, 6) maps to one output (3, 5, 6), so this relation is a function .
(b) {( 4 , 5 ) , ( 4 , 6 ) , ( 5 , 5 ) , ( 5 , 6 )}
The input 4 maps to two different outputs (5 and 6), so this relation is not a function .
(c) {( 6 , 7 ) , ( 6 , 8 ) , ( 7 , 7 ) , ( 7 , 8 )}
The input 6 maps to two different outputs (7 and 8), and the input 7 also maps to two different outputs. So, this relation is not a function .
Evaluate each function at the indicated values:
(a) f ( x ) = 9 − 6 x
f ( − 1 ) = 9 − 6 ( − 1 ) = 9 + 6 = 15
f ( 1 ) = 9 − 6 ( 1 ) = 9 − 6 = 3
f ( − 3 + x ) = 9 − 6 ( − 3 + x ) = 9 + 18 − 6 x = 27 − 6 x
(b) g ( x ) = x 2 − 4 x
g ( 2 ) = ( 2 ) 2 − 4 ( 2 ) = 4 − 8 = − 4
(g(2 - x) = (2 - x)^2 - 4(2 - x) = (4 - 4x + x^2) - (8 - 4x) = x^2 - 4)
Perform operations on functions f ( x ) = 3 x + 4 and g ( x ) = 2 x − 1 :
( f + g ) ( x ) = f ( x ) + g ( x ) = ( 3 x + 4 ) + ( 2 x − 1 ) = 5 x + 3
( f − g ) ( x ) = f ( x ) − g ( x ) = ( 3 x + 4 ) − ( 2 x − 1 ) = x + 5
( f ⋅ g ) ( x ) = f ( x ) ⋅ g ( x ) = ( 3 x + 4 ) ( 2 x − 1 ) = 6 x 2 − 3 x + 8 x − 4 = 6 x 2 + 5 x − 4
( f / g ) ( x ) = g ( x ) f ( x ) = 2 x − 1 3 x + 4 , where g ( x ) = 0 and thus x = 2 1 ensures the denominator is not zero.
