Question #715a8

1 Answer
Jun 5, 2017

(f@g)(x)=3x^2-27

Explanation:

This is a composition of functions. It is essentially a function within a function. The notation states that you are evaluating the inner function, g(x), and then inserting the result into f(x) and re-evaluating.

(f@g)(x)=f(g(x))

Read it as "f of g". I prefer the second notation as you can see which one is inside the other. f(x) is the outer function and g(x) is the inner function.

To evaluate, you replace all the x's of the outer function with the inner function.

f(x)=x -4
g(x)=3x^2-23

(f@g)(x)=f(g(x))=f(3x^2-23)

=(3x^2-23)-4

=3x^2-27

As an example, let's evaluate (f@g)(x) when x=2. You can do this by step by step:

i) Evaluate the inner function

g(2)=3(2)^2-23=-11

ii) Insert this into the outer function and evaluate

f(-11)=-11 -4=-15

But now that we have the composition function, we can insert x=2 directly into it instead to get the same answer:

(f@g)(2)=3(2)^2-27=-15

*As extra but vital information, an important rule for the composition to be defined is this:

"range of inner function " sube " domain of outer function"
I.e.
"ran " g(x) sube "dom "f(x)

This says that the range of g(x) must be a subset or equal to the domain of f(x). If this is not true, you trying to input values into f(x) that are not part of its domain and, thus, cannot be evaluated.
In this case,

"ran " g(x) = [-23,oo)
"dom "f(x)=RR

The below statement is true, so the composition is defined.

[-23,oo) sube RR