Question

Given that f(x)=x^2+5 and g(x)=2x+7, calculate

f∘g(x)= ,
g∘f(x)= ,
f∘f(x)= ,
g∘g(x)

Answer 1

wow which one would you like to do first?

Answer 2

\[f\circ g(x)=f(g(x))=f(2x+7)=(2x+7)^2+5\] then algebra.. do you understand the steps?

Answer 3

satellite what do you mean by algebra isnt that all to it

Answer 4

i can write out the steps in english if you like?

Answer 5

ok

Answer 6

oh i mean probably you would write
\[(2x+7)^2+5=4 x^2+28 x+54\]

Answer 7

OK we want for example
\[g\circ f(x)\] step one, get rid of the circle and write what this really means
\[g(f(x))\]

Answer 8

oh i see

Answer 9

step two,
\[f(x)\] can be any function but now replace it by the specific one you are given. in this case it is
\[f(x)=x^2+5\] so
\[g(f(x))=g(x^2+5)\]

Answer 10

then were you see an "x" in
\[g(x)\] replace it by
\[x^2+5\] using parentheses. in this case you get
\[g(x^2+5)=(x^2+5)+7\]

Answer 11

typo there, should have been
\[g(x^2+5)=2(x^2+5)+7\]

Answer 12

and finally algebra to clean it up

Answer 13

so is it 2x^2 + 10 + 7

Answer 14

well i would say
\[2x^2+17\]

Answer 15

oh ok

Answer 16

where it says g(g) and f(f)...thats strange. ive never seen that before. can you demonstrate it please

Answer 17

it is exactly the same as before.
\[f\circ f(x)=f(f(x))\]

Answer 18

so again start from the inside, and work out
\[f(f(x))=f(x^2+5)=(x^2+5)^2+5\]

Answer 19

the reason this is confusing is because there are so many "x"s
ignore them (mentally) if you can. for example the function
\[g(x_=2x+7\] don't think
"g of x equal two x plus seven"
think
"g means first double, then add seven"

Answer 20

the function is the action, has nothing to do with the variable. so there is no difference between
\[g(x)=2x+7\] and
\[g(z)=2z+7\] and
\[g(\heartsuit)=2\heartsuit+ 7\]