Function Composition

Here is where we introduce a new operation - composition of functions.  With this definition comes new notation.
This new notation reads “f composed with g.”
The idea is to substitute one function into another function.

Given the functions f(x) = 5x – 4 and g(x) = 2x – 1.
Composition of functions is not necessarily a commutative operation, in other words, order matters.

Instructional Video: Composite Functions

Given the functions f(x) = x^2 – 9 and g(x) = x – 3.
Given the functions f(x) = x^2 + 1 and g(x) = sqrt(– 1) where ( x >= 1 ).
At this point we must understand what happens to the domain of a composite function. In the above example it might appear that f o g has a domain of all real numbers.  In fact, the domain is restricted to [1, inf) because that is the domain of g.  The domain of f o consists of all the values in the domain of g that are also in the domain of f.

Given f and g find f o g and state its domain.

YouTube Videos:

Solving Radical Equations

Now that we have learned how to work with radical expressions, we next move on to solving.  Use caution when solving radical equations because the following steps may lead to extraneous solutions, solutions that do not solve the original equation.

Step 1: Isolate the radical.
Step 2: Square both sides of the equation.
Step 3: Solve the resulting equation and then check your answers.

Whenever you raise both sides of an equation to an even power, you introduce the possibility of extraneous solutions so the check is essential here.
The index determines the power to which we raise both sides.  For example, if we have a cube root we will raise both sides to the 3rd power. The property that we are using is
for integers n > 1 and positive real numbers x. After eliminating the radical, we will most likely be left with either a linear or a quadratic equation to solve.
The check mark indicates that we have actually checked that the value is a solution to the equation, do not dismiss this step, it is essential.
Some radical equations have more than one radical expression.  These require us to isolate each remaining radical expression and raise both sides to the nth power until they are all eliminated.  Be patient with these, go slow and avoid short cuts.



Video Examples on YouTube: