Question 1013198
i usually show f composite g(x) as fog(x) and g composite f(x) as gof(x).
as long as it's defined, then you can use it.


so:


f composite g(x) = fog(x) = f(g(x))
g composite f(x) = gof(x) = g(f(x))


problem 1:
Consider f(x)=7x+2, g(x)=x^2.
find fog(x) and gof(x).


problem 2:
Consider f(x)=sqrt{x^2+2}, g(x)=x^2+5.
find fog(x) and gof(x).


we'll do problem 1 first.


f(x) = 7*x + 2
g(x) = x^2


problem 1a.
find fog(x)
fog(x) = f(g(x))
since g(x) = x^2, then you get fog(x) = f(x^2)
the argument of x in f(x) = 7*x + 2 is replaced with x^2 to get:
fog(x) = f(g(x)) = f(x^2) = 7*x^2 + 2.


problem 1b.
find gof(x)
gof(x) = g(f(x))
since f(x) = 7*x + 2, then you get gof(x) = g(7*x + 2)
the argument of x in g(x) = x^2 is replaced with (7*x + 2) to get:
gof(x) = g(f(x)) = g(7*x + 2) = (7*x + 2)^2


we'll do problem 2 next.


f(x)=sqrt{x^2+2}
g(x)=x^2+5.


problem 2a.
find fog(x)
fog(x) = f(g(x))
since g(x) = x^2+5, then you get fog(x) = f(x^2+5)
the argument of x in f(x) = sqrt(x^2 + 2) is replaced with x^2 + 5 to get:
fog(x) = f(g(x)) = f(x^2 + 5) = sqrt((x^2+5)^2 + 2)



problem 2b.
find gof(x)
gof(x) = g(f(x))
since f(x) = sqrt(x^2+2), then you get gof(x) = g(sqrt(x^2 + 2))
the argument of x in g(x) = x^2 + 5 is replaced with sqrt(x^2 + 2) to get:
gof(x) = g(f(x)) = g(sqrt(x^2 + 2) = (sqrt(x^2+2))^2 + 5
the argument of x was replaced with an argument of sqrt(x^2+2).


this might be easier to see if it's hand written.


see the following worksheet.


look below the worksheet to find a reference you might find helpful.


<img src = "http://theo.x10hosting.com/2016/011101.jpg" alt="$$$" </>


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