# Question #715a8

Jun 5, 2017

$\left(f \circ g\right) \left(x\right) = 3 {x}^{2} - 27$

#### 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.

$\left(f \circ g\right) \left(x\right) = f \left(g \left(x\right)\right)$

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

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

$f \left(x\right) = x - 4$
$g \left(x\right) = 3 {x}^{2} - 23$

$\left(f \circ g\right) \left(x\right) = f \left(g \left(x\right)\right) = f \left(3 {x}^{2} - 23\right)$

$= \left(3 {x}^{2} - 23\right) - 4$

$= 3 {x}^{2} - 27$

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

i) Evaluate the inner function

$g \left(2\right) = 3 {\left(2\right)}^{2} - 23 = - 11$

ii) Insert this into the outer function and evaluate

$f \left(- 11\right) = - 11 - 4 = - 15$

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

$\left(f \circ g\right) \left(2\right) = 3 {\left(2\right)}^{2} - 27 = - 15$

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

$\text{range of inner function " sube " domain of outer function}$
I.e.
$\text{ran " g(x) sube "dom } f \left(x\right)$

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

$\text{ran } g \left(x\right) = \left[- 23 , \infty\right)$
$\text{dom } f \left(x\right) = \mathbb{R}$

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

$\left[- 23 , \infty\right) \subseteq \mathbb{R}$