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Mathematics LibreTexts

1.0: Library of functions

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  • Page ID
    10145
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    Thinking out Loud

    What is the meaning of a real-valued function?

    Real Valued functions 

    Definition

    Let \(D \subset \mathbb{R}\). Then \(f:D\to  \mathbb{R}\) is said to be a real valued function from \(D\) to  \( \mathbb{R}\) if for each element \(x \in D\), there exists a unique (one and only one) element  \(y \in  \mathbb{R}\), such that \(y=f(x).\) In this case \( D \) is called domain of \( f \), \(x\) is called an independent variable, and \(y \) is called an dependent variable,

    Note that we will follow the following convention: for a given real valued function \(f\) , the domain of f is denoted by \( D(f) \) and defined by \( D(f) = \) the set of all \(x\) such that  there exists a \(y \in  \mathbb{R}\) such that  \(y=f(x)\).

    Library of functions

    The following examples  of functions will be used in this course:

    Polynomial functions

    Definition

     

    Linear functions

    Example \(\PageIndex{1}\)


    Define \(f: \mathbb{R}\to  \mathbb{R}\) by \(f(x)=x+1\). Then  \( D(f) = \mathbb{R} \).

    x+1.png

    Exercise \(\PageIndex{1}\)

    Define \(f: \mathbb{R}\to  \mathbb{R}\) by \(f(x)=x-1\). Find the domain.

    Answer

    \( D(f) = \mathbb{R} \) 

    x-1.png

    \(\rm I\!R\)

    Quadratic functions 

    Example \(\PageIndex{2}\):

    Define \(f: \mathbb{R}\to  \mathbb{R}\) by \(f(x)=x^2\). Then  \( D(f) = \mathbb{R} \).

    x2.png

    Exercise \(\PageIndex{2}\)

    Define \(f: \mathbb{R}\to  \mathbb{R}\) by \(f(x)=-x^2\). Find the domain.

    Answer
     
    \( D(f) = \mathbb{R} \)
    -x2.png

     Cubic functions

    Example \(\PageIndex{3}\):

    Define \(f: \mathbb{R}\to  \mathbb{R}\) by \(f(x)=x^3\). Then  \( D(f) = \mathbb{R} \).

    .x3.png

    Exercise \(\PageIndex{3}\)

    Define \(f: \mathbb{R}\to  \mathbb{R}\) by \(f(x)=x^3\). Find the Domain

    Answer

    \( D(f) = \mathbb{R} \)

    -x3.png

    Rational functions

    Example \(\PageIndex{4}\)


    Define \(f: \mathbb{R}\to  \mathbb{R}\) by \(f(x)= \displaystyle\frac{x^2-1}{x-1}\). Then  \( D(f)= \mathbb{R} \setminus \{1\} \).

    x.1undefined.png

    Exercise \(\PageIndex{4}\)

    Define \(f: \mathbb{R}\to  \mathbb{R}\) by \(f(x)= \displaystyle\frac{x^2-1}{x+1}\). Find the domain.

    Answer

    \( D(f)= \mathbb{R} \setminus \{-1\}\)

    x.-1undefined.png 

    Example \(\PageIndex{5}\):

    Define \(f: \mathbb{R}\to  \mathbb{R}\) by \(f(x)= \displaystyle\frac{1}{x-1}\). Then  \( D(f)= \mathbb{R} \setminus \{1\} \).

    frac(1)(x-1).png

    Exercise \(\PageIndex{5}\)

    Define \(f: \mathbb{R}\to  \mathbb{R}\) by \(f(x)= \displaystyle\frac{1}{x-1}\). Find the domain.

    Answer

    \( D(f)= \mathbb{R} \setminus \{-1\} \).

    frac(1)(x+1).png

     Irrational functions

    upper half unit circle \(\PageIndex{6}\)


    Define \(f: \mathbb{R}\to  \mathbb{R}\) by \(f(x)=\sqrt{1-x^2}\).  Then  \( D(f)= [-1,1].\) 

    sqrt1-x2.png

    Exercise \(\PageIndex{6}\)

    Define \(f: \mathbb{R}\to  \mathbb{R}\) by \(f(x)=\sqrt{4-x^2}\).  Find the domain.

    Answer

    \( D(f)= [-2,2].\)

    sqrt4-x2.png

     

    Example \(\PageIndex{7}\):

    Define \(f: \mathbb{R}\to  \mathbb{R}\) by \(f(x)=\sqrt{1-x}\).  Then \( D(f)= [0,\infty) \). 

    sqrt{1-x).png

     

    Exercise \(\PageIndex{7}\)

    Define \(f: \mathbb{R}\to  \mathbb{R}\) by \(f(x)=\sqrt{1-x}\). Find the domain.

    Answer

    \( D(f)= [0,\infty) \)

    sqrt{x-1).png

     Piecewise-defined function

    Absolute valued function

    Example \(\PageIndex{8}\)


    Define \(f: \mathbb{R}\to  \mathbb{R}\) by \(f(x)=|x|\).

    Note that \(\sqrt{x^2}=|x|\).

    absolutex.png

    Exercise \(\PageIndex{1}\)

    Define \(f: \mathbb{R}\to  \mathbb{R}\) by \(f(x)=|x|+1\)

    Answer

    absolutex+1.png

    Step functions

    Example \(\PageIndex{9}\): Sign function

    Define \(f: \mathbb{R\setminus\{0\}}\to  \mathbb{R}\) by \(f(x)=\frac{|x|}{x}\).

     

    Example \(\PageIndex{10}\): Unit Step function

    Define \(f: \mathbb{R}\to  \mathbb{R}\) by \(f(x)=\).

    Contributors