**Video Transcript**

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Transcript
**Special Functions**

We’re here in the Dr. Frank N. Stein Laboratory in Europe where Dr. Stein’s understudy, Dr. Em Kay, is taking her final exam in reanimation. Dr. Em Kay has to understand the **graphs** of **special functions** in order to make her specimen, Scarion, perform a few unassisted actions in order to pass her class.

### Graphs of Special Functions

To do this, she can use the **graphs of special functions**! Dr. Kay has to input the equation of a function into the EKG and then shock Scarion with her new defibrillator, the Tesla DeFib 18. Dr. Em Kay knows that all of the **equations** she should use have to be **functions**, which means that they all have to pass the **vertical line test**. The good doctor has just finished her first trial. For this, she input the special function **f(x) = |x|**. She knows that the **domain** is **all real numbers** and the **range** is all numbers **greater than or equal** to 0. Okay, this move seems to work. Scarion's arms are in a 'v' shape, just like an **absolute value graph**. Dr. Em Kay hits the abort button so she can test another function.

Dr. Kay tries again. This time, she chooses **f(x) = 1/x** as her function. The domain for this function is all real numbers, except x = 0 and the range is all real numbers, except y = 0. Graphs that have **variables** in the **denominator** usually have **asymptotes** in their graphs. An asymptote is an **invisible lin**e such that another line or curve may get closer and closer to, but will never reach. This function has a **vertical asymptote** at x = 0 and a **horizontal asymptote** at y = 0. This looks like a fun dance move! Dr. Em Kay is ecstatic!

After resetting Scarion one more time, Dr. Kay moves on to the next **formula**. The next function Dr. Kay tries is **sqrt(x)**. The domain for this function is x ≥ 0 and the range is y ≥ 0. Using this graph, Dr. Kay can turn Scarion into a smooth operator! Dr. Kay will soon be certified to perform reanimations!

For the next move, Dr. Kay wants Scarion to be able to do an arm wave. Dr. Kay tries the **sine function**, **f(x) = sin x** for Scarion's right arm and **g(x) = cos x** for Scarion's left arm. The domain and range for the **sine** and **cosine functions** are the same: the domain for both functions is all the real numbers and the range is -1 ≤ f(x) ≤ 1. Arm wave complete! There's just one last move Scarion needs to learn.

The last function Dr. Kay enters into her machine is **f(x) = tan x**. The domain and range for **tangent**: the domain for tangent is all the real numbers that are **not divisible** by π/2 + π(n) and the range is all the real numbers. Look at all the asymptotes on the tangent graph! Whoa! Look at that! *Dance, Scarion, dance!!!*

### Summary

Let's do a quick review of the functions we covered in this video. The functions Dr. Kay used are: the **absolute value of 'x'**, **one over 'x'**, the **square root of 'x'** and the basic **trig functions**: **sine**, **cosine** and **tangent**. The domain of the absolute value of 'x' as well as all the **trig functions** is all real numbers. Tangent, however, has one exception, shown here, where 'n' is any **integer value**. The function one over 'x' has an exception at x = 0 since dividing by 0 is undefined. Otherwise, the domain of the function one over 'x' is all real numbers. Our final domain, for the square root of 'x', is all numbers greater than or equal to 0.

For our ranges, we have two functions that will produce values greater than or equal to 0 - the absolute value function and the square root function. The function one over 'x' has a range of all real numbers except f(x) = 0. For the trig functions, both sine and cosine produce numbers between -1 and 1, inclusive. And finally, the tangent function has a range of all the real numbers. Let's check in on the good doctor, mmmkay?

Look at Scarion move!! wait...some of those dance moves weren't programmed by Dr. Kay! *Scarion...she's ALIVE!!!*

1 commentWow... That's really helpful.