# MST124 Unit 3 - Functions

# Introduction

Whenever one quantity depends on another, we say that the first quantity is a **function** of the second quantity.

# 1 Functions and their graphs

## 1.1 Sets of real numbers

A **set** is a collection of objects. The objects could be anything at all.
An **element** or **member** is each of the objects in a set.
Elements of the set **belong to** or are **in** a set.

A set can be specified by enclosing a list of elements in curly brackets, e.g.: $$ S = {5, 12, 34, 58} $$

A set can be described, e.g.: Let $T$ be the set of all odd integers. Sets are usually denoted with capital letters.

**Empty set** is denoted with $
\emptyset$.
$\in$ means is in and $\notin$ means is not in.

An **intersection** is a process of forming new set out of overlapping members of two or more sets and is denoted as $A \cap B$.

**Union** is a process of forming new set out of all the members of two or more sets and is denoted as $A \cup B$.

$A$ is a **subset** of $B$ when every element of $A$ is also an element of $B$. A subset is denoted as $A \subseteq B$.
Every set is a subset of itself and the empty set is a subset of every set.

The set of all real number is denoted as $\mathbb{R}$.

An **interval** is a set of real numbers that correspond to a continuous section of number line.

An **endpoint** is a number that lies at the end of an interval.
The whole set of real numbers $\mathbb{R}$ is an interval with no endpoints.

**Closed** is such interval that includes all of its endpoints.
**Open** interval doesn't include any of its endpoints.
**Half-open** interval includes only one of its endpoints.

$\mathbb{R}$ doesn't have endpoints, so it's both open and closed.

**Inequality signs** can be used to describe intervals.

**Interval notation** can also describe intervals e.g.:
$$
4 \le x < 7
$$
is denoted as:
$$
[4, 7)
$$

An interval that extends indefinitely is denoted by $\infty$ or $-\infty$.

Union of intervals can be described as e.g.: $$ (-\infty, 2] \cup [4, 6) $$

## 1.2 What is a function?

A **function** consists of:

- a set of allowed input values, called the
**domain**of the function - a set of values in which every output value lies, called the
**codomain**of the function - a process, called the
**rule**of the function, for converting each input value into*exactly one*output value.

A function can be denoted with **function notation**, e.g.:
$$
h(x) = x^2
$$

The **image set** is a set of values in the codomain that *do* occur as output values.

Every value in the domain of a function must have *exactly one* output value.

If $f$ is a function, and $x$ is any value in its domain, then the value $f(x)$ is called the **image of** $x$ **under** $f$, or the **value of** $f$ **at** $x$, or $f$ **maps** $x$ to $f$.

**Real functions** have domain and codomain consisting of real numbers.

## 1.3 Specifying functions

Function specification consists of its domain and rule, e.g.: $$ f(x) = x^{2}+ 1 ~~ (0 \le x \le 6) $$ or

$$ f(x) = x^{2}+ 1 ~~ (x \in [0, 6]) $$

When a function is specified by *just a rule*, it's understood that the domain of the function is the largest possible set of real numbers for which the rule is applicable.

## 1.4 Graphs of functions

is the set of points $(x, y)$ in the coordinate plane such that $x$ is in the domain of $f$ and $y = f(x)$.

### Functions increasing or decreasing on an interval

A function $f$ is **increasing on the interval** $I$ if for all values $x_1$ and $x_2$ in $I$ such that $x_1 < x_2$,
$$
f(x_1) < f(x_2)
$$

A function $f$ is **decreasing on the interval** $I$ if for all values $x_1$ and $x_2$ in $I$ such that $x_1 < x_2$,
$$
f(x_1) > f(x_2)
$$
(The interval $I$ must be part of the domain of $f$.)

## 1.5 Image sets of functions

In the same way as a domain of function can be visualized on the $x$-axis the image set of a function can be visualized on the $y$-axis.

The image set consists of all the possible output numbers, that is, all the points on the $y$-axis that lie directly to the right or left of a point on the graph.

## 1.6 Some standard types of functions

**Linear functions**- has the rule of the form $f(x) = mx + c$.**Constant function**- a linear function whose rule is of the form $f(x) = c$.**Quadratic functions**- has the rule of the form $f(x) = ax^{2} + bx + c$.

### Polynomial functions

Linear and quadratic functions are particular types of polynomial functions. Other example of polynomial function is: $$ f(x) = 2x^{4} - 5x^{3} + x^{2} + 2x - 3 $$

In general, a **polynomial expression in $x$** is any expression that is a sum of finitely many terms, each of which is of the form $ax^{n}$ where $a$ is a number and $n$ is a non-negative integer.

**Degree** of a polynomial expression or function is the highest power of the variable $x$ in it.

Quadratic functions are polynomial functions of degree 2.

Linear functions are polynomial functions of degree 1, 0 or no degree at all.

Polynomial functions of degrees 3, 4, and 5 are called

**cubic****quartic****quintic**

functions, respectively

Every polynomial function (with domain $\mathbb{R}$) that isn't a constant function **tends to infinity** or **tends to minus infinity** at the left and right.

**Dominant term** is a term in a rule of function with the highest power of $x$ and it defines whether the function tends to infinity or minus infinity. It's called dominant because it outweights (*dominates*) the sum of the values taken by all other temrs for large values of $x$.

If the dominant term has a plus sign and

- contains an even power of $x$, then the graph tends to infinity at both ends.
- contains an odd power of $x$, then the graph tends to minus infinity at the left and to infinity at the right.

If the dominant term has a minus sign and

- contains an even power of $x$, then the graph tends to minus infinity at both ends.
- contains an odd power of $x$, then the graph tends to infinity at the left and to minus infinity at the right.

### The modulus function

The **modulus** of a real number (or **magnitude**, **absolute value**) is its 'distance from zero' and for a real number $x$ its denoted as $|x|$.

THe modulus function is $$ f(x) = |x| $$

It follows form the definition that $$ \begin{equation*} |x| = \begin{cases} x \quad &\text{if} , x \ge 0 \ -x \quad &\text{if} , x < 0 \ \end{cases} \end{equation*} $$

So the graph of the modulus function is the same as the graph of $y = x$ when $x \ge 0$ and same as the graph of $y = -x$ when $x < 0$. It has a corner at the origin and image set of $[0, \infty)$.

### The reciprocal function

If $x$ is any non-zero number, then the **reciprocal** of $x$ is $1/x$.
The **reciprocal function** is
$$
f(x) = \frac{1}{x}
$$
With the domain of $(-\infty, 0) \cup (0, \infty)$.

It forms following graph

If a curve has a property when it gets *arbitrarily close* to straight line as it gets further away from origin then the straight line is called the **asymptote** of the curve.

Hence the coordinate axes are asymptotes of the graph of the reciprocal function.

### Rational functions

The reciprocal and all polynomial functions are particular examples of *rational functions*.

The **rational function** is a function whose rule is of the form
$$
f(x) = \frac{p(x)}{q(x)}
$$
where $p$ and $q$ are polynomial functions.

*f* is polynomial function when $q$ is a constant function.
*f* is a reciprocal function when $p(x) = 1$ and $q(x) = x$.

# 2 New functions from old functions

## 2.1 Translating the graphs of functions

A **translation** of a shape means sliding it to different position without rotating or reflecting it.

Suppose that $f$ is a function and $c$ is a constant. To obtain the graph of:

- $y = f(x) + c$, translate the graph of $y = f(x)$ up by $c$ units (the translation is down if $c$ is negative)
- $y = f(x - c)$, translate the graph of $y = f(x)$ to the right by $c$ units (the translation is to the left if $c$ is negative).

A translations can be combined. An equation $y = f(x)$ can be moved $c$ units to the right by replacing $x$ as $y = f(x - c)$. A shift up can be achieved by adding a constant $d$: $$ y = f(x - c) + d $$

Hence the original equation is translated to the right by $c$ units and up by $d$ units. Order of operations doesn't matter.

## 2.2 Scaling the graphs of functions vertically

**Constant multiple** is a function obtained by multiplying the RHS of a function rule by a constant, e.g. $g(x) = 8x^3$ is a constant multiple of the function $f(x)= x^3$.

The resulting effect on the graph of function is called the **vertical scaling**.

#### Vertical scaling of graphs

Suppose that $c$ is a constant. To obtain the graph of $y = cf(x)$, scale the graph of $y = f(x)$ vertically by a factor of $c$.

**Negative** of a function $f$ is the result of multiplying the right-hand side of the rule of a function $f$ by $-1$.

Vertical scaling can be combined with vertical and/or horizontal translations of graphs, e.g. $y = 4x^{3}+ 1$ scales the equation $y = x^3$ vertically by $4$ units, then translates it up by $1$ unit.
The order of operations *does* often matter.

## 2.3 Scaling the graphs of functions horizontally

The horizontal scaling is achieved by replacing each occurrence of the input variable $x$ in the right-hand side of the rule of a function by an expression of the form $x/c$, where $c$ is constant.

The resulting effect on the graph of a function is called **horizontal scaling**.

#### Horizontal scalings of graphs

Suppose that $c$ is a non-zero constant. To obtain the graph of

$$ y = f(\frac{x}{c}) $$

scale the graph of $y = f(x)$ horizontally by a factor of $c$.

### Reflections of graphs in the coordinate axes

To obtain the graph of $y = -f(x)$, reflect the graph of $y = f(x)$ in the $x$-axis. To obtain the graph of $y = f(-x)$, reflect the graph of $y = f(x)$ in the $y$-axis.

# 3 More new functions from old functions

## 3.1 Sums, differences, products and quotients of functions

Suppose that $f$ and $g$ are functions.

- The
**sum**has the rule $h(x) = f(x) + g(x)$ - There are two
**differences**- $h(x) = f(x) - g(x)$
- $h(x) = g(x) - f(g)$

- The
**product**has the rule $h(x) = f(x)g(x)$ - There are two
**quotients**- $h(x) = \frac{f(x)}{g(x)}$
- $h(x) = \frac{g(x)}{f(x)}$

## 3.2 Composite functions

Suppose $f$ and $g$, if the output of $f$ lies in the domain of $g$ it can be input to it.

A **composite** function is function whose rule is given by this process and whose domain is the largest set of real numbers to which you can apply the process.
It's denoted by $g \circ f$ (the function that is applied *first* is *second* in the notation).

#### Definition of composite functions

Suppose that $f$ and $g$ are functions. The **composite function** $g \circ f$ is the function whose rule is
$$
(g \circ f)(x) = g(f(x))
$$
and whose domain consists of all the values $x$ in the domain of $f$ such that $f(x)$ is in the domain of $g$.

The process of forming a composite function from two functions is called **composing** the functions.
Functions $f$ and $g$ have two composite functions

- $g \circ f$, meaning apply $f$ first, than $g$
- $f \circ g$, meaning apply $g$ first, than $f$

## 3.3 Inverse functions

The inverse function of a function $f$, which is denoted by $f^{-1}$, is the function whose mapping diagram is obtained by reversing the direction of all the arrows in this full version of diagram.

#### One-to-one functions

A function $f$ is **one-to-one** if fo all values $x_1$ and $x_2$ in its domain such that $x_1 \ne x_2$,
$$
f(x_1) \ne f(x_2)
$$

Only one-to-one functions have inverse functions.
**Invertible** function is a function that has an inverse function.

#### Definition of inverse functions

Suppose that $f$ is one-to-one function, with domain $A$ and image set $B$. Then the **inverse function**, or simply **inverse**, of $f$, denoted by $f^{-1}$, is the function with domain $B$ whose rule is given by
$$
f^{-1}(y) = x, ~~~~~\text{where} ~~ f(x) = y
$$
The image set of $f^{-1}$ is $A$.

#### Strategy to find the rule of the inverse function of a one-to-one function $f$

- Write $y = f(x)$ and rearrange this equation to express $x$ in terms of $y$.
- Use the resulting equation $x = f^{-1}(y)$ to write down the rule of $f^{-1}$. (Usually, change the input variable from $y$ to $x$.)

If a function is either increasing on its whole domain, or decreasing on its whole domain, then it is one-to-one and so has an inverse function.

For any pair of inverse functions $f$ and $f^{-1}$,

- $(f^{-1} \circ f)(x) = x$, for every value $x$ in the domain of $f$, and
- $(f \circ f^{-1})(x) = x$, for every value $x$ in the domain of $f^{-1}$

### Graphs of inverse functions

The graphs of a pair of inverse functions are the reflections of each other in the line $y = x$ (when the coordinate axes have equal scales).

### Functions that aren't one-to-one

Starting with a function $f$ that is not one-to-one, specify a new function that has same rule as $f$, but a smaller domain with following conditions satisfied:

- the new function is one-to-one and therefore has an inverse
- the image set of the new function is the same as the image set of the original function

E.g. for $f(x) = x^{2}$ the new function can be $g(x) = x^{2} ~~ (x \in [0, \infty))$. $g$ is one-to-one with inverse $g^{-1}(x) = \sqrt{x}$.

A function like $g$ that is obtained from another function $f$ by removing some numbers from its domain and keeping the rule is called a **restriction** of the original function $f$.

# 4 Exponential functions and logarithms

## 4.1 Exponential functions

Is a function whose rule is of the form
$$
f(x) = b^x
$$
where $b$ is a positive constant, not equal to 1.
$b$ is called the **base number** of the exponential function.

#### Graphs of exponential functions

The graph of the function $f(x) = b^{x}$, where $b > 0$ and $b \ne 1$, has the following features.

- The graph lies entirely above the $x$-axis.
- If $b > 1$, then the graph is increasing, and it gets steeper as $x$ increases.
- If $0 < b < 1$, then the graph is decreasing, and it gets less steep as $x$ increases.
- The $x$-axis is an asymptote.
- The $y$-intercept is 1.
- The closer the value of $b$ is to 1, the flatter is the graph.

### The exponential function

The graph of every exponential function $f(x) = b^{x}$ passes through the point $(0, 1)$. And its steepness depends on the value of $b$.

The exponential function with rule $f(x) = e^{x}$ has the special property that its gradient is exactly 1 at the point $(0,1)$, this function is referred to as ** the exponential function**.

## 4.2 What is a logarithm?

Logarithms have a particular *base*
Logs to base 10 are known as **common logarithms** with following definition.
The **logarithm to base** 10 of a number $x$ is the power to which 10 must be raised to give the number $x$. e.g.:
$$
\begin{align}
\log_{10}100 = 2 \\[0.6em]
\log_{10}1000 = 3 \\[0.6em]
\log_{10}(\frac{1}{10})= -1
\end{align}
$$

#### General definition of logarithms

The **logarithm to base** $b$ of a number $x$, denoted by $\log_bx$, is the power to which the base $b$ must be raised to give the number $x$. So the two equations
$$
y = \log_bx ~~~ \text{and} ~~~ x = b^y
$$
are equivalent.
Remeber that:

- the base $b$ must be positive and not equal to 1
- only positive numbers have logarithms, but logarithms themselves can be any number.

#### Logarithm of the number 1 and logarithm of the base

For any base $b$, $$ \log_b1 = 0 ~~~ \text{and} ~~~ \log_bb = 1 $$

The most common bases of logarithms are $2$, $10$ and $e$.
Logarithms to the base $e$ are called **natural logarithms** and the notation $\ln$ is often used.

#### Natural logarithms

The **natural logarithm** of a number $x$, denoted by $\ln x$, is the power to which the base $e$ must be raised to give the number $x$. So the two equations
$$
y = \ln x ~~~\text{and} ~~~ x = e^y
$$
are equivalent.
Natural logarithms share following properties with common logs
$$
\ln 1 = 0 ~~~ \text{and} ~~~ \ln e = 1
$$

## 4.3 Logarithmic functions

Are functions with a rule of the form $$ f(x) = \log_b x $$ where $b$ is a positive constant, not equal to 1.

The logarithmic function $f(x) = \log_b x$ is the inverse function of the exponential function $g(x) = b^x$ since $$ y = \log_bx ~~~ \text{and} ~~~ x = b^y $$ are rearrangements of each other. Hence the graphs of both functions are reflections of each other in the line $y = x$.

#### Graphs of logarithmic functions

The graph of the function $f(x) = \log_bx$, where $b > 0$ and $b \ne 1$, has the following features.

- The graph lies entirely to the right of the $y$-axis.
- If $b > 1$, then the graph is increasing, and it gets less steep as $x$ increases.
- If $0 < b < 1$, then the graph is decreasing, and it gets less steep as $x$ increases.
- The $y$-axis is an asymptote.
- The $x$-intercept is 1.
- The closer the value of $b$ is to 1, the steeper is the graph.

For any base $b$, $$ \log_b(b^{x})= x ~~~ \text{and} ~~~ b^{\log_bx} = x $$ In particular, $$ \ln(e^{x}) = x ~~~ \text{and} ~~~ e^{\ln x} = x $$ (A composition of $f(x) = b^{x}$ and $f^{-1}(x) = \log_bx$)

## 4.4 Logarithm laws

#### Three logarithm laws

$$ \begin{aligned} \log_bx + \log_by = \log_b(xy) \\[0.5em] \log_bx - \log_by = \log_b(\frac{x}{y}) \\[0.5em] r\log_bx = log_b(x^r) \end{aligned} $$

### Solving exponential equations

An exponential equation has the unknown in the exponent, e.g. $$ 5^{x}= 130 $$

And the third logarithm law can be used to solve those. It's useful to swap the sides for this purpose: $$ \log_b(x^{r)}= r\log_b(x) ~~~ \text{in particular} ~~~ \ln(x^{r}) = r\ln(x) $$

## 4.5 Alternative form for exponential functions

Any exponential function $f(x) = b^x$, where $b$ is a positive constant not equal to 1, can be written in the alternative form $$ f(x) = e^{kx} $$ where $k$ is a non-zero constant. The constant $k$ is given by $k = \ln b$.

This fact gives following alternative definition of an exponential function
An **exponential function** is a function whose rule is of the form
$$
f(x) = e^{kx}
$$
where $k$ is a non-zero constant.
And properties of exponential graphs can be rewritten as

#### Graphs of exponential functions

The graph of the function $f(x) = e^{kx}$, where $k \ne 0$, has the following features.

- The graph lies entirely above the $x$-axis.
- If $k > 0$, then the graph is increasing, and it gets steeper as $x$ increases.
- If $k < 0$, then the graph is decreasing, and it gets less steep as $x$ increases.
- The $x$-axis is an asymptote.
- The $y$-intercept is 1.
- The closer the value of $k$ is to 0, the flatter is the graph.

The graph of every exponential function is a horizontal scaling of the graph of the exponential function $f(x) = e^x$.

## 4.6 Exponential models

Are defined by a function with rule of the form $f(x) = ab^{x}$. That is equal to $f(x) = ae^{kx}$ in the alternative form.

$f(x) = ae^{kx}$ is obtained by

- vertically scaling the graph of the function $g(x) = e^{kx}$ by the factor of $a$.

$g(x) = e^{kx}$ is obtained by

- horizontally scaling the graph of the function $h(x) = e^{kx}$ by a factor of $1/k$.

Hence the graph of the function $f(x) = ae^{kx}$ is obtained by scaling the graph of the function $h(x) = e^x$ both horizontally and vertically.

A quantity is **changed exponentially** when the change can be modeled by a function with rule of the the form $f(x) = ae^{kx}$.

If $a$ and $k$ are positive then the quantity is said to **grow exponentially** and the function is called **exponential growth function**.

If $a$ is positive but $k$ is negative then it's said the quantity **decays exponentially**, the function is **exponential decay function** and graph is called an **exponential decay curve**.

#### A characteristic property of exponential growth and decay functions

If $f(x) = ae^{kx}$, then whenever $p$ units are added to the value of $x$, the value of $f(x)$ is multiplied by $e^{kp}$

The reason is following. Say the value of $x$ changes from $$ f(x) = ae^{kx} ~~~ \text{to} ~~~ f(x + p) = ae^{k(x + p)} $$ Then $$ ae^{k(x + p)} = a^{kx + kp} = ae^{kx}e^{kp} = f(x) \times e^{kp} $$

The factor by which $f(x)$ is multiplied is called **growth** or **decay** factor.

- Growth factors are $> 1$
- Decay factors are between 0 and 1, exclusive.

#### Doubling and halving periods

Suppose that $f$ is an exponential growth function. Then $p$ is the **doubling period** of $f$ if whenever you add $p$ to $x$, the value of $f(x)$ doubles.

Similarly, suppose that $f$ is an exponential decay function. Then $p$ is the **halving period** of $f$ if whenever you add $p$ to $x$, the value of $f(x)$ halves.

#### Strategy to find a doubling or halving period

If $f(x) = ae^{kx}$ is an exponential growth function (so $k > 0$), then the doubling period of $f$ is the solution $p$ of the equation $e^{kp} = 2$; that is, $p = (\ln2)/k$.

Similarly, if $f(x) = ae^{kx}$ is an exponential decay function (so $k < 0$), then the halving period of $f$ is the solution $p$ of the equation $e^{kp} = \frac{1}{2}$; that is, $p = \ln \frac{1}{2}/k= -(\ln 2)/k$.

# 5 Inequalities

## 5.2 Rearranging inequalities

Carrying out any of the following operations on an inequality gives an equivalent inequality.

- Rearrange the expressions on one or both sides.
- Swap the sides,
*provided you reverse the inequality sign*. - Do any of the following things to both sides:
- add or subtract something
- multiply or divide by something that's positive
- multiply or divide by something that's negative,
*provided you reverse the inequality sign*.

Both sides shouldn't be multiplied by a variable unless it's limited o positive values only.