Rules of Exponents
Since we often see exponents throughout all math courses, it is important to understand the rules of exponents. We need to understand how to distribute, add, multiply and divide exponents in order to simplify expressions or manipulate equations that have exponents. The rules of exponents, like those involving multiplication of terms, are important to learn and will be used throughout Algebra I and II and Calculus.
Negative Exponents
It's important to understand what It means to have negative exponents and zero exponents. Negative exponents put the exponentiated term in the denominator of a fraction and zero exponents just make the term equal to one. We can use negative exponents for cancelling with positive exponents while solving equations or simplifying expressions, although we need to keep in mind the rules of multiplying exponents.
Simplifying Expressions with Exponents
We can use what we know about exponents rules in order to simplify expressions with exponents. When simplifying expressions with exponents we use the rules for multiplying and dividing exponents, and negative and zero exponents. In Algebra and higher math courses such as Calculus, we will often encounter simplifying expressions with exponents.
One to One Function
After learning the definition of a function, we can extend it to define a one to one function. A one to one function has not only one output for every input, but also only one input in the domain for every output in therange. Another interesting type is an invertible function, or a function that has an inverse. The graph of a one to one or invertible function has unique and interesting characteristics.
Finding the Inverse
Once we learn the definition of a function's inverse we learn how to find the algebraic inverse, or how to find the inverse using algebraic methods. There are different methods for finding the inverse, the most common of which is to switch the dependent and independent variables and solve for the dependent variable. This is an important step in learning how to prove the inverse of a function.
Graphing Inverse Functions
In order to understand graphing inverse functions, students should review the definition of inverse functions, how to find the inverse algebraically and how to prove inverse functions. The graphs of inverse functions and invertible functions have unique characteristics that involve domain and range. Techniques for graphing inverse functions can make it easier to graph certain functions by hand.
Multiplying Exponents
There are different rules to follow when multiplying exponents and when dividing exponents. If we are multiplying similar bases, we simply add the exponents. If we are dividing, we simply subtract the exponents. If an exponent is outside the parentheses, it is distributed to the inside terms. It's important to understand the rules of multiplying exponents so that we can simplify expressions with exponents.
Scientific Notation
In the applied sciences, scientific notation is often employed as a method of notation for ease of writing and reading. When dealing with real world situations, the numbers we get as solutions are rarely whole numbers and scientific notation gives us rules to follow when using ugly numbers that have a lot of decimal places. In order to understand scientific notation, we must also have a solid understanding of exponents.
Inverse, Exponential and Logarithmic Functions
Inverse, Exponential and Logarithmic Functions teaches students about three of the more commonly used functions, and uses problems to help students practice how to interpret and use them algebraically and graphically. Students can learn the properties and rules of these functions and how to use them in real world applications through word problems such as those involving compound interest and exponential growth and decay that they will find on their homework.
Definition of Inverse
The definition of a function can be extended to define the definition of inverse of a function. Along with one to one functions, invertible functions are an important type of function. The definition of inverse says that a function's inverse switches its domain and range. The definition of inverse helps students to understand the unique characteristics of the graphs of invertible functions.
Proving Inverse Functions
The definition of a function can be extended to define the definition of an inverse, or an invertible function. It's important to understand proving inverse functions, and the method of proving inverse functions helps students to better understand how to find inverse functions. Students should review how to find an inverse algebraically and the basics of proofs.
Graphing Exponential Functions
There are certain functions, such as exponential functions, that have many applications to the real world and have useful inverse functions. Graphing exponential functions is used frequently, we often hear of situations that have exponential growth or exponential decay. The inverses of exponential functions are logarithmic functions. The graphs of exponential functions are used to analyze and interpret data.
Solving Exponential Equations
In the real world we often hear terms like exponential growth or exponential decay, when discussing solving exponential equations such as those used in compounding interest problems. In order to understand solving exponential equations, students should understand the significance of exponential functions and logarithmic functions.
