RADICAL FUNCTIONS

First thing to know about radical functions is that it involves a radical with a variable in the radicand.


One major rule to keep in mind is that the x-value in the radicand can be any number, as long as it is not negative. Taking the square, cube, fourth, or any root of a negative number is impossible.

Now let's look at a basic radical function.

y = √x

An important thing to do that people tend to forget is to solve for the inequality. Take the value from inside the radical and equate it to zero.

x = 0

Now simply replace the symbol with the inequality symbol.

x ≥ 0

From that, we know that x has to be greater or equal to zero.

The simplest way to represent this radical function on a graph is to use a table of values. There are other methods, such as comparing graphs and solving the equation, but table of values seem to be the easiest method. To make it even easier for yourself, you should pick values for x that you can take the square root out of.

Examples: Values of 0, 1, 4, 9, and 16




 
Irrational numbers such as 3, 5, 6, 7 and so on can also be chosen as the x-value, but the result will turn out with decimals, making it harder for yourself to graph.










Now comes the graphing. Simply plot these points down.





 



It will turn out looking like this!





 


Looking at the graph will help you figure out the domain, range, x-intercept, and y-intercept. However, from solving the inequality earlier, you have already figured out the domain.

Inequality: x ≥ 0
Domain: { x | x ≥ 0, XER }

Range: { y | y ≥ 0, YER}

X-intercept: 0
(Point that crosses x-axis)

Y-intercept: 0
(Point that crosses y-axis)

Natural Base (e) and Natural Logarithm (ln)

Hi guys! Here’s a short review on Natural Logarithms.

• Natural Base

The natural base ‘e’ is another strange number frequently used by  mathematicians which approximately equals to 2.71828

2.718281828x = ex

• Natural Logarithm

Natural Logarithms comes from the fancy Latin name Logarithmus naturali , giving the abbreviation ln. Natural Logarithms is really just the inverse of ‘e’.

e^x = y  —>  lne y = x

or

e^y = x   —>  ln x = y

Understanding Natural Logarithms

Reminder:  All Natural Logarithms use the same patterns and laws as Common Logarithms.

Expanding Natural Logarithmic Expressions

Roots—> Fraction Exponents Division—> Subtraction

Multiplication —> Addition Exponents —> Move to the front

Expand.

Follow expansion rules.

Do not forget to apply negative through.

Simplifying Natural Logarithmic Expressions

Rearrange terms (all negative terms  must be on the end) Fraction Exponents—> Roots

Addition —> Multiplication Move to the front —> Exponents Subtraction —> Division

Simplify.

Rearrange.

Do not forget to FACTOR OUT the negative.

Follow the simplification rules.

Solving Natural Logarithmic Equations

Simplify Convert Logarithm into exponential form (use 7 rule) Solve for x or any missing variable Check answers by plugging answer(s) back into the equation.

Place one logarithm with base e on each side of the eq’n. Using the Laws of Logarithms turn the exponent into a coefficient to the front of its logarithm. If the exponent contains a binomial, use brackets around the exponent. If the exponent is in brackets, multiply through brackets. Collect logarithms on one side and isolate the variable on the other. If there is more than one term containing the missing variable isolate the variable. Solve for the variable with your calculator Check your answer.

Evaluate. Round answer to three decimal place

Follow simplification rules. Rearrange.

Simplify further.

Convert to exponential form using 7 rule.

Solve for x.

Solving Using Cancellation Property

Evaluate

Take the left and right hand side of the equation and raise it as a power of e.

Cancel out same bases.

Raise left and right hand side as a power of e one last time.

Cancel out same bases.

Exponential and Logarithmic Functions Recap

Hello. Here’s a quick re-cap of what we have learned about exponential and logarithm functions before winter break.

Graphing Exponential Functions

Key points to remember:

•   Basic curve: y=c^x
• The same transformation rules apply:

 y=(a)c^b(x-h)+k 

o   a – vertical stretch by a factor of lal about the x axis.

-          ex: y= (5)c^x (Vertical stretch by a factor of 5 about the x axis.)

o   b – horizontal stretch by a factor of 1/b about the y axis.

-          ex. y=c^2x (Horizontal stretch by a factor of ½ about the y axis.)

o   h – horizontal translation

-          ex. y=c^(x+2) (Horizontal translation by 2 units to the left)

o   k – vertical translation

-          ex. y=c + 3 (Vertical translation by 3 units up)

 

Steps:

1.  Determine the horizontal asymptote. (The horizontal asymptote is related to the vertical translation, k.)
2.  Table of Values; pick x values.
3.   Plug in the x values into the equation in order to find the corresponding y values.
4.  Graph.

y = 2^z

HA @ y=0

Table of Values:

         

Graph:

Graphing Logarithmic Functions

Key points to remember:

•   A logarithm function is the inverse of an exponential function.

Steps

1.  Convert logarithmic function to exponential function using the 7 Rule.
2.   Determine the vertical asymptote.
3. Table of Values; pick y values.
4. Plug in the y values into the equation in order to find the corresponding x values.
5.  Graph.

f(x) = log₂x -> 2^y = x

VA @ x=0

Table of Values

          

Graph:

Expanding and Simplifying Logarithmic Expressions

Key points to remember when expanding:

•   Roots -> fraction exponent
•  Division -> subtraction
•  Multiplication -> addition
•  Exponent -> coefficient

Ex.

 

Key points to remember when simplifying:

•  Move all negative terms to the end
• Coefficient -> exponent
•  Addition -> multiplication
•  Subtraction -> division
•  Fraction exponent -> roots