Functions come in all different shapes. A few are very closely related and others are very different, but often confused. For example, what is the difference between \begin{align*}x^2\end{align*}

#### Watch This

http://www.youtube.com/watch?v=3a7UbMJpeIM Khan Academy: Graphing a Quadratic Function

http://www.youtube.com/watch?v=Mi6OJ4TAufY Khan Academy: Graphing Radical Functions

http://www.youtube.com/watch?v=9SOSfRNCQZQ Khan Academy: Graphing Exponential Functions

http://www.youtube.com/watch?v=outcfkh69U0 Khan Academy: Graphs of Square Root Functions

#### Guidance

If mathematicians are cooks, then families of functions are their ingredients. Each family of functions has its own flavor and personality. Before you learn to combine functions to create an infinite number of potential models, you need to get a clear idea of the name of each function family and how it acts.

**The Identity Function: \begin{align*}f(x)=x\end{align*} f(x)=x**

The identity function is the simplest function and all straight lines are transformations of the identity function family.

**The Squaring Function: \begin{align*}f(x)=x^2\end{align*} f(x)=x2**

The squaring function is commonly called a parabola and is useful for modeling the motion of falling objects. All parabolas are transformations of this squaring function.

**The Cubing Function: \begin{align*}f(x)=x^3\end{align*} f(x)=x3**

The cubing function has a different kind of symmetry than the squaring function. Since volume is measured in cubic units, many physics applications use the cubic function.

**The Square Root Function: \begin{align*}f(x)=\sqrt{x}=x^{\frac{1}{2}}\end{align*} f(x)=x√=x12**

The square root function is not defined over all real numbers. It introduces the possibility of complex numbers and is also closely related to the squaring function.

**The Reciprocal Function: \begin{align*}f(x)=x^{-1}=\frac{1}{x}\end{align*} f(x)=x−1=1x**

The reciprocal function is also known as a hyperbola and a rational function. It has two parts that are disconnected and is not defined at zero. Simple electric circuits are modeled with the reciprocal function.

So far all the functions can be grouped together into an even larger function family called the power function family.

**The Power Function Family: \begin{align*}f(x)=cx^a\end{align*} f(x)=cxa**

The power function family has two parameters. The parameter \begin{align*}c\end{align*}

**The Exponential Function Family: \begin{align*}f(x)=e^x\end{align*} f(x)=ex**

The exponential function family is one of the first functions you see where \begin{align*}x\end{align*}

**The Logarithm Function: \begin{align*}f(x)=\ln x\end{align*} f(x)=lnx**

The log function is closely related to the exponential function family. Many people confuse the graph of the log function with the square root function. Careful analysis will show several important differences. The log function is the basis for the Richter Scale which is how earthquakes are measured.

**The Periodic Function Family: \begin{align*}f(x)=\sin x\end{align*} f(x)=sinx**

The sine graph is one of many periodic functions. Periodic refers to the fact that the sine wave repeats a cycle every period of time. Periodic functions are extremely important for modeling tides and other real world phenomena.

**The Absolute Value Function: \begin{align*}f(x)=|x|\end{align*} f(x)=|x|**

The absolute value function is one of the few basic functions that is not totally smooth.

**The Logistic Function: \begin{align*}f(x)=\frac{1}{1+e^{-x}}\end{align*} f(x)=11+e−x**

The logistic function is a combination of the exponential function and the reciprocal function. This curve is very powerful because it models population growths where the maximum population is limited by environmental resources.

**Example A**

Compare and contrast the graphs of the two functions: \begin{align*}f(x)=\ln x\end{align*}

**Solution: **

*Similarities:* Both functions increase without bound as \begin{align*}x\end{align*}

*Differences:* The log function approaches negative infinity as \begin{align*}x\end{align*}

**Example B**

Describe the symmetry among the function families discussed in this concept. Consider both reflection symmetry and rotation symmetry.

**Solution: **

Some function families have reflective symmetry with themselves:

\begin{align*}y=x, y=x^2, y=\frac{1}{x}, y=|x|\end{align*}

Some function families are rotationally symmetric:

\begin{align*}y=x, y=x^3, y=\frac{1}{x},y=\sin x, y=\frac{1}{1+e^{-x}}\end{align*}

Some pairs of function families are full or partial reflections of other function families:

\begin{align*}y = x^2, y=\sqrt{x}\end{align*}

\begin{align*}y = e^x, y=\ln x\end{align*}

**Example C**

Which function families are unbounded above and below?

**Solution:**

Look for the function families that don’t have an overall maximum or minimum value.

\begin{align*}y=x, y=x^3, y=\frac{1}{x},y= \ln x\end{align*}

**Concept Problem Revisited**

While \begin{align*}x^2\end{align*}

#### Vocabulary

A ** function family** is a group of functions that all have the same basic shape.

A ** parameter** is a constant embedded in a function that affects the shape of the function in a limited and specific way.

** Unbounded above** means that the function gets bigger than any specific number you can choose.

** Unbounded below** means that the function can get smaller than any specific number you can choose.

** Continuous** means that the function can be drawn entirely without lifting your pencil.

#### Guided Practice

1. Which functions are discontinuous?

2. Which functions always have a positive slope over the entire real line?

3. Which functions are defined for all \begin{align*}x\end{align*}

**Answers:**

1. The reciprocal function is the only function included here that is discontinuous.

2. \begin{align*}y=x, y=e^x, y=\frac{1}{1+e^{-x}}\end{align*}

3. \begin{align*}y=x, y=x^2, y=x^3, y=e^x, y=\sin x, y=|x|, y=\frac{1}{1+e^{-x}}\end{align*}

#### Practice

For 1-10, sketch a graph of the function from memory.

1. \begin{align*}y = e^x\end{align*}

2. \begin{align*}y= \ln (x)\end{align*}

3. \begin{align*}y= \sin (x)\end{align*}

4. \begin{align*}y= x^2\end{align*}

5. \begin{align*}y= |x|\end{align*}

6. \begin{align*}y = \frac{1}{x}\end{align*}

7. \begin{align*}y = \frac{1}{1+e^{ - x}}\end{align*}

8. \begin{align*}y = \sqrt{x}\end{align*}

9. \begin{align*}y = x^3\end{align*}

10. \begin{align*}y = x\end{align*}

11. Which function is not defined at 0? Why?

12. Which functions are bounded below but not above?

13. What are the differences between \begin{align*}y = x^2\end{align*} and \begin{align*}y = x^3\end{align*}?

14. What is a similarity between \begin{align*}y = e^x\end{align*} and \begin{align*}y = \ln (x)\end{align*}?

15. Explain why \begin{align*}y = \sqrt{x}\end{align*} is not defined for all values of \begin{align*}x\end{align*}.