1.3.1a Composite Function Example 1

Example 1
If

f : x \mapsto x + 5

, and

g : x \mapsto x^{2} + 2 x + 3

, find
(a) the value of gf(2),
(b) the value of fg(2)
(c) the composite functions fg
(d) the composite functions gf
(e) the composite functions

g^{2}

(f) the composite functions

f^{2}

\begin{array}{l} f g (x) = f (x^{2} + 2 x + 3) \\ = (x^{2} + 2 x + 3) + 5 \\ = x^{2} + 2 x + 3 + 5 \\ = x^{2} + 2 x + 8 \end{array}

correction for part (e)

\begin{array}{l} g^{2} (x) = g g (x) \\ = g (x^{2} + 2 x + 3) \\ = {(x^{2} + 2 x + 3)}^{2} + 2 (x^{2} + 2 x + 3) + 3 \\ = x^{4} + 2 x^{3} + 3 x^{2} + 2 x^{3} + 4 x^{2} + 6 x \\ + 3 x^{2} + 6 x + 9 \\ = x^{4} + 4 x^{3} + 10 x^{2} + 12 x + 9 \end{array}

Maria

April 19, 2018 at 2:43 pm | Reply

hiii sorry i dont understand where the 5 came for question C for composite functions
- myhometuition
  
  April 21, 2018 at 8:11 am | Reply
  
  Thanks for pointing out our mistake, correction had been made accordingly.
ARIZAF

September 11, 2020 at 7:36 am | Reply

i dont understand calculation in (e).. is it the final answer or has extra calculation.. i get x^4 + 4x^3+12x^2+16x+15