Chapter 5 Real functions

5.1 Real Functions

5.1.1 Definitions

Real function: a collection of pairs of real numbers \((x, y)\), such that each number \(x\) has at most one image \(y\).

Notation: \(f\), \(g\), \(h\), \(f_1\), \(f_2\), ...

Terminology: \((x,y) \in \mathbb {R}^2\)

• • \(x\) argument, independent variable, abscissa, \(x\)-value

• • \(y\) image, dependent variable, ordinate, \(y\)-value

• • \(\in \) element of

• • \(\mathbb {R}^2\) set of all pairs of real numbers

Representations:

• • function rule: \(f : \mathbb {R} \to \mathbb {R}:x \mapsto f(x)\)

• • function value table

• • graph

5.1.2 Characteristics of Real Functions

Domain

Definition: the set of all \(x\)-values for which the function has an image.

Notation: \(\text {dom } f\)

Graphically:

Range

Definition: the set of all images of a function.

Notation: \(\text {ran } f\)

Graphically:

Zeroes

Definition: an \(x\)-value for which the image is equal to zero.

Algebraic: Set \(y=0\) in \(y=f(x)\). So search for all \(x\) for which

\[f(x)=0\]

Graphically:

Sign Behavior

Definition: determine the sign (positive, negative, or zero) of the image \(y\) as the argument \(x\) goes through the domain.

Method: Sign diagram with \(+\), \(-\), \(0\) and \(|\)

Increasing and Decreasing

Definitions:

• • \(f\) is increasing in an interval \(I\) if for the \(x\)-values \(a,b \in I\) with \(a < b\) it holds that

  \[f(a) \leq f(b)\]

• • \(f\) is decreasing in an interval \(I\) if for the \(x\)-values \(a,b \in I\) with \(a < b\) it holds that

  \[f(a) \geq f(b)\]

• • \(f\) that is both increasing and decreasing in an interval \(I\) is called constant on that interval \(I\)

Method: Diagram with \(\nearrow \) and \(\searrow \)

Extrema

Definitions:

• • real number \(m\) is the minimum of \(f\) on an interval \(I\) if for all \(x\in I:\)

  \[f(x)\geq m\;.\]

• • real number \(M\) is the maximum of \(f\) on an interval \(I\) if for all \(x\in I:\)

  \[f(x)\leq M\;.\]

Method: Diagram with \(\nearrow \) and \(\searrow \)

5.2 Polynomials

5.2.1 Polynomial

A polynomial is a sum of monomials:

\begin{align*} P(x) &= a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0\\ &= \sum _{i=0}^{n} a_i x^i \end{align*}

• • \(x\): variable

• • \(a_i\in \mathbb {R}\): coefficients

• • degree \(deg(P(x))\): the highest exponent of \(x\) in \(P\)

• • value of \(P(x)\) for a given \(a\): replace \(x\) with \(a\)

• • root: all \(a\) for which \(P(a)=0\)

5.2.2 Degree after operations

\begin{align*} deg(A(x) + B(x)) &\leq \max (deg(A(x)),deg(B(x)))\\ deg(A(x) \cdot B(x)) &= deg(A(x)) + deg(B(x)) \end{align*}

5.2.3 Euclidean Division

\[A(x) = D(x) \cdot Q(x) + R(x)\]

\[ \qquad \qquad \text { with } deg(R(x)) \lt deg(D(x)) \vee R(x)=0\]

• • \(A(x)\) : dividend

• • \(D(x) \neq 0\) : divisor

• • \(Q(x)\) : quotient

• • \(R(x)\) : remainder

• • exact division: \(R(x)=0\)

• • non-exact division: \(R(x) \neq 0\)

• • \(deg(A(x)) \lt deg(D(x)) \quad \Rightarrow \quad Q(x)=0 \wedge R(x)=A(x)\)

• • \(Q(x)\) and \(R(x)\) exist and are unique

5.3 Polynomial Functions

5.3.1 General

Polynomial function: a real function with function formula

\[f(x)= a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\]

Degree: \(n\)-th degree

5.3.2 Polynomial Functions of Degree 0

• • Synonym: constant functions

• • Form:

  \[f(x) = a \qquad (a \in \mathbb {R})\]

• • Graph: horizontal line parallel to the \(x\)-axis.

• • Domain: \(\mathbb {R}\)

• • Range: \(\{a\}\)

• • Intersection with \(y\)-axis: \((0, a)\)

• • Zeros:

  \[ \begin {cases} \text {none} & (a \neq 0) \\ \text {every } x \in \mathbb {R} & (a = 0) \end {cases} \]

• • Sign diagram:

  \[ \begin {array}{c|lcr} x & -\infty & & +\infty \\ \hline f(x) & & \text {sign of }a & \end {array} \]

5.3.3 Polynomial Functions of Degree 1

• • Synonyms: first-degree functions, linear functions

• • Form:

  \[f(x) = ax+b \qquad (a \in \mathbb {R}_0, b \in \mathbb {R})\]

• • Graph: oblique line

  \[ \begin {cases} \text {increasing oblique line} \nearrow & (a \gt 0) \\ \text {decreasing oblique line} \searrow & (a \lt 0) \end {cases} \]

• • Gradient: \(a\) slope / rate of change

• • Domain: \(\mathbb {R}\)

• • Range: \(\mathbb {R}\)

• • Zero: \(x_0=-\dfrac {b}{a}\)

• • Intersection with \(x\)-axis: \((-\dfrac {b}{a}, 0)\).

• • Intersection with \(y\)-axis: \((0, b)\)

• • Sign diagram:

  \[ \begin {array}{c|lcccr} x & -\infty & & -\dfrac {b}{a} & & +\infty \\[0.2cm] \hline f(x) & & \substack {\text {opposite }\\\text {sign } a } & 0 & \substack {\text {sign }\\a} & \end {array} \]

• • Extrema: none

5.3.4 Polynomial Functions of Degree 2

• • Synonyms: second-degree functions, quadratic functions

• • Form:

  \[f(x) = ax^2+bx+c \qquad (a \in \mathbb {R}_0, b,c \in \mathbb {R})\]

• • Domain: \(\mathbb {R}\).

• • Range:

  \[ \begin {cases} \text {ran }f = [f(-\dfrac {b}{2a}), +\infty ) & (a \gt 0)\\ \text {ran }f = (-\infty , f(-\dfrac {b}{2a})] & (a \lt 0) \end {cases} \]

• • Zeros: \(ax^2+bx+c=0\)

  • – \(D=b^2-4ac\)

  • – \(D>0 \Rightarrow \) two distinct zeros

    \[x_1=\dfrac {-b-\sqrt {D}}{2a} \qquad x_2=\dfrac {-b+\sqrt {D}}{2a}\]

  • – \(D=0 \Rightarrow \) one zero

    \[x_0=-\dfrac {b}{2a}\]

  • – \(D<0 \Rightarrow \) no zeros

• • Sign diagram:

  • – \(D>0\)

    \[ \begin {array}{c|ccccc} x & & x_1 & & x_2 &\\ \hline f(x) & \substack {\text {sign}\\a} & 0 & \substack {\text {opposite}\\\text {sign }a} & 0 & \substack {\text {sign}\\a} \end {array} \]

  • – \(D=0\)

    \[ \begin {array}{c|lcccr} x & -\infty & & x_0 & & +\infty \\ \hline f(x) & & \substack {\text {sign}\\a} & 0 & \substack {\text {sign}\\a} & \end {array} \]

  • – \(D<0\)

    \[ \begin {array}{c|lcr} x & -\infty & & +\infty \\ \hline f(x) & & \text {sign } a & \end {array} \]

• • Graph: parabola with symmetry axis parallel to the \(y\)-axis

  • – \(a>0 \Rightarrow \) upward-opening parabola (concave up / happy)

  • – \(a<0 \Rightarrow \) downward-opening parabola (concave down / unhappy)

• • Vertex:

  \[(-\dfrac {b}{2a}, f(-\dfrac {b}{2a}))\]

  \[ = (-\dfrac {b}{2a}, -\dfrac {D}{4a})\]

  with \(D=b^2-4ac\)

• • Increasing and decreasing:

  • – \(a \lt 0\)

    \[ \begin {array}{c|lcr} x & & -\dfrac {b}{2a} & \\[0.2cm] \hline f(x) & \nearrow & \underset {=f(-\frac {b}{2a})}{\text {max}} & \searrow \end {array} \]

  • – \(a \gt 0\)

    \[ \begin {array}{c|lcr} x & & -\dfrac {b}{2a} & \\[0.2cm] \hline f(x) & \searrow & \underset {=f(-\frac {b}{2a})}{\text {min}} & \nearrow \end {array} \]

5.4 Rational functions

5.4.1 Basic function

Form:

\[f(x)=\dfrac {1}{x}\]

Graph: Hyperbola

Domain:

\[\text {dom }f = \mathbb {R}_0\]

Range:

\[\text {ran }f = \mathbb {R}_0\]

Zeros:

\[\text {zeros } f: \diagup \]

Asymptotes:

\[\text {H.A.}: y=0\]

\[\text {V.A.}: x=0\]

5.4.2 Homographic functions

Form:

\[f(x)=\dfrac {ax+b}{cx+d} \qquad (c\neq 0 \wedge a \cdot d \neq b \cdot c)\]

Discuss:

• • Function rule: \(f(x)=\dfrac {ax+b}{cx+d} \;.\)

• • poles \(f\): \(-\dfrac {d}{c}\)

• • \(\text {dom }f = \mathbb {R} \setminus \{\text {poles}\}\)

• • zeros \(f\): \(\begin {cases} x_0=-\dfrac {b}{a} & (a\neq 0)\\ \text {none} & (a=0) \end {cases}\)

• • V.A.: \(x=\) poles

• • H.A.: \(y=\dfrac {\text {coeff }x}{\text {coeff }x}\)

• • \(\text {ran }f = \mathbb {R} \setminus \{\text {H.A.}\}\)

• • Sign diagram: by splitting numerator and denominator

• • Determine intersections

• • Sketch graph

• • Increasing and decreasing: based on the graph

5.4.3 Rational functions

Form:

\[y=\frac {t(x)}{n(x)}\]

• • \(t(x)\): numerator, polynomial function

• • \(n(x)\): denominator, polynomial function

5.5 Exponential Functions

5.5.1 Exponential Functions

Form:

\[f(x) = a^x \qquad (a \in \mathbb {R}^+_0\setminus \{1\})\]

• • \(a\): base

• • \(x\): exponent

Characteristics

• • Domain: \(\mathbb {R}\)

• • Range: \(\mathbb {R}^+_0\)

• • Zeroes: none

• • Special points: \((0,1),\; (1,a)\)

Graph

• • \(0 \lt a \lt 1\)

  

• • \(a \gt 1\)

  

5.6 Logarithmic Functions

5.6.1 Logarithmic Function

Form:

\[f(x) = \log _a x\]

• • \(a\): base \(a\in \mathbb {R}^+_0\setminus \{1\}\)

• • \(x\): argument

Characteristics:

• • \(\text {dom }\log _a x = \mathbb {R}^+_0\)

• • \(\text {ran }\log _a x = \mathbb {R}\)

• • Special points: \((1, 0),\;(a, 1)\)

• • \(a \gt 1\)

  • – \(\log _a x\) is increasing in \(\mathbb {R}^+_0\)

  • – \(\lim _{+\infty } \log _a x = +\infty \)

  • – \(\lim _{x\underset {>}{\to }0} \log _a x = -\infty \)

• • \(0 \lt a \lt 1\)

  • – \(\log _a x\) is decreasing in \(\mathbb {R}^+_0\)

  • – \(\lim _{+\infty } \log _a x = -\infty \)

  • – \(\lim _{x\underset {>}{\to }0} \log _a x = +\infty \)

• • Asymptotes:

  • – V.A.: \(y\)-axis with equation \(x=0\)

Graph:

5.6.2 Main Property of Logarithmic Functions

Property: The logarithmic function with base \(a\) and the exponential function with base \(a\) are inverse functions.

In symbols:

\[f(x)=a^x \Rightarrow f^{-1}(x)=\log _ax\]

Graphically: The graphs of the logarithmic and exponential functions are mirror images of each other with respect to the first bisector in an orthonormal coordinate system.

• • \(a \gt 1\)

  

• • \(0 \lt a \lt 1\)

  

5.7 Trigonometric Functions

5.7.1 Sine Function

Form:

\[f(x) = a \sin (bx+c) + d\]

Characteristics:

• • Amplitude: \(|a|\)

• • Period: \(T = \dfrac {2\pi }{|b|}\)

• • Phase Shift: \(-\dfrac {c}{b}\)

• • Vertical Shift: \(d\)

• • Domain: \(\mathbb {R}\)

• • Range: \([d - |a|, d + |a|]\)

Graph:

5.7.2 Cosine Function

Form:

\[f(x) = a \cos (bx+c) + d\]

Characteristics:

• • Amplitude: \(|a|\)

• • Period: \(T = \dfrac {2\pi }{|b|}\)

• • Phase Shift: \(-\dfrac {c}{b}\)

• • Vertical Shift: \(d\)

• • Domain: \(\mathbb {R}\)

• • Range: \([d - |a|, d + |a|]\)

Graph:

5.7.3 Tangent Function

Form:

\[f(x) = a \tan (bx+c) + d\]

Characteristics:

• • Period: \(T = \dfrac {\pi }{|b|}\)

• • Phase Shift: \(-\dfrac {c}{b}\)

• • Vertical Shift: \(d\)

• • Domain: \(\mathbb {R} \setminus \left \{ \dfrac {\pi }{2} + k\pi : k \in \mathbb {Z} \right \}\)

• • Range: \((-\infty , +\infty )\)

Graph: