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### How to Verify the Inverse of a Function

At times, your textbook or teacher may ask you to verify that two given functions are actually inverses of each other. To do this, you need to show that both [more…]

### Understanding the Properties and Identities of Logs

You need to know several properties of logs in order to solve equations that contain them. Each of these properties applies to any base, including the common and natural logs: [more…]

### How to Solve Logarithmic Equations

Logarithmic equations take different forms. As a result, before solving equations that contain logs, you need to be familiar with the following four types of log equations: [more…]

### How to Identify Even and Odd Functions and their Graphs

Knowing whether a function is even or odd helps you to graph it because that information tells you which half of the points you have to graph. These types of functions are symmetrical, so whatever is on [more…]

### How to Graph Parent Functions

In mathematics, you see certain graphs over and over again. For that reason, these original, common functions are called *parent graphs,* and they include graphs of quadratic functions, square roots, absolute [more…]

### Understanding the Rules of Exponential Functions

Exponential functions follow all the rules of functions. However, because they also make up their own unique family, they have their own subset of rules. The following list outlines some basic rules that [more…]

### How to Graph and Transform an Exponential Function

Graphing an exponential function is helpful when you want to visually analyze the function. Doing so allows you to really see the growth or decay of what you’re dealing with. The basic parent function [more…]

### How to Operate on Functions

In your pre-calculus class, you may be asked to operate with two or more functions. Operating with functions can involve addition, subtraction, multiplication, or division. [more…]

### How to Graph Parent Functions and Transformed Logs

Want some good news, free of charge? Graphing parent functions and transformed logs is a snap! You can change any log into an exponential expression, so this step comes first. You then graph the exponential [more…]

### How to Graph a Rational Function with Denominator Having the Higher Degree

After you calculate all the asymptotes and the *x-* and *y-*intercepts for a rational function, you have all the information you need to start graphing the function. In any rational function where the denominator [more…]

### How to Calculate Outputs for Rational Functions

In pre-calculus, you can calculate outputs for rational functions. A *rational function* is a function that can be expressed as the quotient of two polynomials, such that [more…]

### Apply the Impulse Function to Circuit Analysis

The impulse function, also known as a Dirac delta function, helps you measure a spike that occurs in one instant of time. Think of the spiked impulse function [more…]

### Apply the Unit Step Function to Circuit Analysis

The unit step (Heavyside) function models the behavior of a switch (off/on). The unit step function can describe sudden changes in current or voltage in a circuit. The unit step function looks like, well [more…]

### Apply the Exponential Function to Circuit Analysis

The *exponential function* is a step function whose amplitude *V** _{k}* gradually decreases to 0. Exponential functions are important to circuit analysis because they’re solutions to many problems in which a circuit [more…]

### Sinusoidal Functions and Circuit Analysis

The sinusoidal functions (sine and cosine) appear everywhere, and they play an important role in circuit analysis. The sinusoidal functions provide a good approximation for describing a circuit’s input [more…]

### How to Combine Various Transformations

Certain mathematical expressions allow you to combine stretching, shrinking, translating, and reflecting a function all into one graph. An expression that shows all the transformations in one is [more…]

### How to Find the Critical Numbers for a Function

All local extrema occur at critical points of a function — that's where the derivative is zero or undefined (but don't forget that critical points aren't always local extrema). So, the first step in finding [more…]

### How to Determine Maximum Acceleration of a Moving Object

Acceleration is the derivative of velocity. If a function gives the position of something as a function of time, you differentiate the position function to get the velocity function, and you differentiate [more…]