Problem 1 Adding and Subtracting Functions

Let f(x) = 4x + 7 and g ( x ) = x + x . g , open x close , equals square root of x plus x .  What are f+ g and f − g ? f minus g question mark  What are their domains?

( f + g ) ( x ) = f ( x ) + g ( x ) = ( 4 x + 7 ) + ( x + x ) = 5 x + x + 7 ( f − g ) ( x ) = f ( x ) − g ( x ) = ( 4 x + 7 ) − ( x + x ) = 3 x − x + 7 table with 2 rows and 1 column , row1 column 1 , open , f plus g , close . open x close , equals f , open x close , plus g , open x close , equals . open , 4 x plus 7 , close . plus . open , square root of x plus x , close . equals 5 x plus square root of x plus 7 , row2 column 1 , open , f minus g , close . open x close , equals f , open x close , minus g , open x close , equals . open , 4 x plus 7 , close . minus . open , square root of x plus x , close . equals 3 x minus square root of x plus 7 , end table

The domain of f is the set of all real numbers. The domain of g is all x ≥ 0 . x greater than or equal to 0 .  The domain of both f + g and f − g f minus g  is the set of numbers common to the domains of both f and g, which is all x ≥ 0 . x greater than or equal to 0 .

✓ Got It?

• Let f ( x ) = 2 x 2 + 8 f open x close equals , 2 x squared , plus 8  and g ( x ) = x − 3 . g open x close equals x minus 3 .  What are f + g and f − g ? f minus g question mark  What are their domains?

Problem 2 Multiplying and Dividing Functions

Let f ( x ) = x 2 − 9 f open x close equals , x squared , minus 9  and g(x) = x + 3. What are f • g and f g f over g  and their domains?

( f ⋅ g ) ( x ) = f ( x ) ⋅ g ( x ) = ( x 2 − 9 ) ( x + 3 )   = x 3 + 3 x 2 − 9 x − 27 ( f g ) ( x ) = f ( x ) g ( x ) = x 2 − 9 x + 3 = ( x − 3 ) ( x + 3 ) x + 3 = x − 3 , x ≠ − 3 table with 3 rows and 2 columns , row1 column 1 , open , f dot g , close . open x close , equals f , open x close , dot g , open x close , column 2 equals . open . x squared , minus 9 . close . open , x plus 3 , close , row2 column 1 , , column 2 equals , x cubed , plus 3 , x squared , minus 9 x minus 27 , row3 column 1 , open , f over g , close . open x close , equals . fraction f , open x close , over g , open x close end fraction . equals . fraction x squared , minus 9 , over x plus 3 end fraction , column 2 equals . fraction open , x minus 3 , close . open , x plus 3 , close , over x plus 3 end fraction . equals x minus 3 comma x not equal to negative 3 , end table

The domain of both f and g is the set of real numbers, so the domain of f ⋅ g f dot g  is also the set of real numbers.

The domain of f g f over g  is the set of all real numbers except x ≠ − 3 , x not equal to negative 3 comma  because g ( − 3 ) = 0 . g open negative 3 close equals 0 .

The definition of f g f over g  requires that you consider the zero denominator in the original expression for f ( x ) g ( x ) fraction f , open x close , over g , open x close end fraction  despite the fact that the simplified form has the domain all real numbers.

✓ Got It?

• Let f ( x ) = 3 x 2 − 11 x − 4 f open x close equals , 3 x squared , minus 11 x minus 4  and g ( x ) = 3 x + 1 . g open x close equals 3 x plus 1 .  What are f ⋅ g f dot g  and f g f over g  and their domains?