Composition of Functions - Pre-Calculus

f(g(x)) simply means: where ever you see an x in the equation f(x), replace it with g(x).

So, doing just that, we get

,

which simplifies to

.

Since

our simplified expression becomes,

.

g(f(x)) simply means replacing every x in g(x) with f(x).

After simplifying, it becomes

When doing a composition of functions such as this one, you must always remember to start with the innermost parentheses and work backward towards the outside.

So, to begin, we have

.

Now we move outward, getting

.

Finally, we move outward one more time, getting

.

The composite function means to plug in the function of into the function for every x value in the function.

Therefore the composition function becomes:

.

The composite function means to plug in the function into for every x value.

Therefore the composite function becomes,

Solve for the value of .

Solve for the value of .

Solve for the value .

To find the composite function we start from the most inner portion of the expression and work our way out.

The composite funtion means to replace every entry x in f(x) with the entire function g(x).

The composite function notation means to swap the function into for every value of . Therefore:

The composite function notation means to swap the function into for every value of . Therefore:

The composite function notation means to swap the function into for every value of . Therefore:

Working inside out, first do .

This is,

.

Now we will do .

This is

Working from the inside out, first we will find a function for .

This is:

, which we can simplify slightly to .

Now we will plug this new function into the function k:

.

Since ln is the inverse of e to any power, this simplifies to .

To find simply substiute for every x in and solve.

Let's discuss what the problem is asking us to solve. The expression (read as as "f of g of x") is the same as . In other words, we need to substitute into .

Substitute the equation of for the variable in the given function:

Next we need to FOIL the squared term and simplify:

FOIL means that we multiply terms in the following order: first, outer, inner, and last.

First:

Outer:

Inner:

Last:

When we combine like terms, we get the following:

Substitute this back into the equation and continue to simplify.

None of the answers are correct.

"G of F of X" means substitute f(x) for the variable in g(x).

Foil the squared term and simplify:

First:

Outer:

Inner:

Last:

So

Now evaluate the composite function at the indicated value of x:

Replace and substitute the value of into so that we are finding .

Given and , find .

Begin by breaking this into steps. I will begin by computing the step, because that will make the late steps much more manageable.

Next, take our answer to and plug it into .

So we are close to our final answer, but we still need to multiply by 3.

Making our answer 84.

and .

To find we plug in the function everywhere there is a variable in the function .

This is the composition of "f of g of x".

Foil the square and simplify: