Question

For f(x)=9x+5 and g(x)=x² find the following composite functions and state the domain of each

a) f o g b) g o f c) f o f d) g o g

a. the domain of f o g is

b. (g o f )(x)=

a. the domain of g o f is

c. ( f o f )(x)=

the domain of f o f is

d. ( g o g)(x)=

the domain of g o g is

Answer 1

f(x)=9x+5 and g(x)=x²

a) fog

= f( g(x) ) [ g(x) = x² ]

= f ( x² ) [ f(x) = 9x + 5 . replace x with x²]

= 9x² + 5

Therefore fog(x) =9x² + 5

Domain: The   domain of a function is the set of input or argument values for which the function id real and defined.

(i.e The domain is the set of all possible x-values which will make the function "work", and will output real y-values) The domain has no undefined points nor domain constraints. Therefore , the domain of fog = 9x² + 5 is . for all real numbers. In Interval Notation

b) (g o f )(x)

g ( f(x) )

= g( 9x+5 )

= ( 9x + 5 )² [ g(x) = x² so replace x with 9x + 5  ]

= (9x)² + 25 + 2 (9x)(5)   [ ( a + b )² = a² + b² + 2 (a)(b) ]

= 81x² + 25 + 90x

Therefore the domain gof is

. Interval Notation

c)

( f o f )(x)= f ( f(x) )

= f ( 9x + 5 )

= 9(9x + 5 ) + 5

= 81x + 45 + 5

= 81x + 50

The domain of fof is

. Interval Notation

d)

( g o g)(x)

= g ( g(x) )

= g ( x² )

= (x²)²

= x⁴

Therefore (g o g)(x) = x⁴

The function has no undefined points nor domain constraints.

so the domain of gog is

. Interval Notation