Special Functions – Practice Problems

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Not all functions are linear. The graphs of nonlinear functions are not lines, and so are sometimes a bit trickier to see. We will take a look at some special nonlinear functions to help us get a better intuition for how nonlinear functions can behave.

The first of such special function we will look at is
f(x) = |x| (absolute value of x).
Its domain consists of all real numbers and its range is all positive real numbers.
The graph has two opposite diagonal lines above the x-axis.

Next we have
f(x) = ⅟x (one over x).
Its domain is all real numbers except zero and its range is also all real numbers except zero.
Its graph is two outward curves that approach but never touch the x-axis and the y-axis.

The third special function is
f(x) = √x (square root of x).
The domain and range of this function both include all positive real numbers, including zero.
Its graph is a curve that goes outward to the right (in the first quadrant of the coordinate system)..

We will also look at
f(x) = sin x and f(x) = cos x (sine of x and cosine of x)
Both of these functions’ domains include all real numbers, while both of their ranges will have real numbers between, and including, negative one and positive one. Both graphs look like waves that are found between the lines y=1 and y=-1.

Finally we have:
f(x) = tan x (tangent of x).
The domain is all real numbers, except values of π n+ π⁄2, where n is any integer, and the range is all real numbers.
Its graph is multiple curves that approach but never touch all values of x of the form π n+ π⁄2.

Graphing nonlinear functions can be complicated! But it’s definitely a bit easier to draw a function’s graph when you know more about its domain and range; watching this video on graphing special functions will surely convince you.

Use functions to model relationships between quantities.


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Exercises in this Practice Problem
Decide which graph belongs to which function.
Write the definition of an asymptote.
Determine the corresponding function equation.
Explain the expressions' domain and range.
Describe how to draw the graph of a function.
Examine the domain, range, and asymptotes of the functions.