Showing posts with label mathematics. Show all posts
Showing posts with label mathematics. Show all posts

Tuesday, November 20, 2012

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.
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Monday, November 19, 2012

Solving Logarithmic Equations

Use the one-to-one property for logarithms to solve logarithmic equations. If we are given an equation with a logarithm of the same base on both sides we may simply equate the arguments.

   Step 1: Use the rules of exponents to isolate a logarithmic expression (with the same base) on both sides of the equation.
   Step 2: Set the arguments equal to each other.
   Step 3: Solve the resulting equation.
   Step 4: Check your answers.
 
Be sure to check to see if the solutions that you obtain solve the original logarithmic equation. In this study guide we will put a check mark next to the solution after we determine that it really does solve the equation. This process sometimes results in extraneous solutions so we must check our answers.
Solve.

Of course, equations like these are very special.  Most of the problems that we will encounter will not have a logarithm on both sides. The steps for solving them follow.

   Step 1: Use the properties of the logarithm to isolate the log on one side.
   Step 2: Apply the definition of the logarithm and rewrite it as an exponential equation.
   Step 3: Solve the resulting equation.
   Step 4: Check your answers.
If the answer to the logarithmic equation makes the argument negative then it is extraneous.  This does not preclude negative answers. We must be sure to check all of our solutions.

Instructional Video: Solving Logarithmic Equations

Solve.
Tip:  Not all negative solutions are extraneous!  Look at the previous set of problems and see that some have negative answers. The check mark indicates that we actually plugged the answers in to see that they do indeed solve the original.  Please do not skip this step, extraneous solutions occur often.
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