Functions and Mappings

Mappings

Consider the function $f$ defined by $f : x \mapsto 2x + 1$. If we input the value $2$ for $x$, $f$ gives $5$ as output, and we say that this function maps $2$ to $5$, written $2 \mapsto 5$. Similarly $f$ maps $-2$ to $-3$, $-1$ to $-1$ and $0$ to $1$. Here one value of the independent variable $x$ maps to just one value of the dependent variable $f(x)$, and no value of $f(x)$ is reached from two different values of $x$.

Now consider $f : x \mapsto x^2 - 2x + 6$. This function maps $-1$ to $9$, $0$ to $6$, $1$ to $5$, $2$ to $6$ and $3$ to $9$. One value of $x$ again gives just one value of $f(x)$, but a value of $f(x)$ is not necessarily obtained from only one value of $x$: both $0$ and $2$ map to $6$.

So $x \mapsto 2x + 1$ is a one-one mapping and $x \mapsto x^2 - 2x + 6$ a many-one mapping.

We may regard a function as a rule for mapping a number $a$ to a number $b$. The strict meaning of the word, however, is restricted to those relationships in which one input value gives rise to just one output value, and a function in the sense of Definition 2.1 has this built in, since it associates to each number a single other number. Both mappings above are functions. The relationship $x \mapsto \pm\sqrt{x}$ is not: input $4$ and it gives two outputs, $2$ and $-2$. It is one-many, a perfectly good mapping for positive values of $x$, and not a function.

Graphs of Functions

Take again $f(x) = x^2 - 2x + 6$. For any chosen value of $x$ the corresponding value of $f(x)$ can be calculated.

Arranged in order of increasing $x$, the values of $f(x)$ show a pattern, and the pattern is easier to see when the results are displayed graphically: we plot the values of $f(x)$ on a vertical number line against the corresponding values of $x$ on a horizontal one.

Figure 3.2 was drawn through values of $x$ one unit apart. Values taken from a larger range, or closer together, all lie on the same smooth curve, so the curve represents every pair of related values of $x$ and $f(x)$. From it we can read off properties of the function.

1. The lowest point on the curve is where $x = 1$ and $f(x) = 5$. Although $x$, as the independent variable, may be given any real value, the corresponding value of $f(x)$ is always greater than $5$, or equal to $5$ when $x = 1$. We say that the function has a least value of $5$.
2. The curve is symmetrical about the line $x = 1$: two values of $x$ equidistant from $1$, of the form $1 + a$ and $1 - a$, give the same value of $f(x)$.
3. Conversely, every value of $f(x)$ above the least corresponds to two values of $x$, of the form $1 + a$ and $1 - a$.

All three can be confirmed algebraically. Completing the square on the right-hand side gives

For the first, $(x - 1)^2$ is a squared quantity and so is never negative. Its least value is $0$, which occurs when $x = 1$, so the least value that $f(x)$ can have is $5$, at $x = 1$.

For the second, take two values of $x$ placed symmetrically on either side of $x = 1$:

so values of $x$ symmetrical about $x = 1$ give the same value of $f(x)$.

For the third, the values of $x$ giving any particular value $c$ of $f(x)$ solve $x^2 - 2x + 6 - c = 0$, and by the quadratic formula the roots of this equation are

For these to be real we need $c \geqslant 5$, which is the least value once more, and the two values of $x$ corresponding to one value of $f(x)$ are symmetrical about $x = 1$.

Domain and Range

For $f(x) = x^2 - 2x + 6$ we have assumed that any real value of $x$ may be input. A full definition of a function must state the set of permissible input values, so the function illustrated above is fully defined as

where $x \in \RR$ means that $x$ may be any real number.

The domain is not always the whole of $\RR$, as the examples below show. The analysis above also showed that, for the domain $x \in \RR$, the output values were limited: the range of $f : x \mapsto x^2 - 2x + 6$, $x \in \RR$, is $f(x) \geqslant 5$.

The Quadratic Function

Functions of similar form usually have properties in common, so their graphs are usually similar in shape. Knowing the common characteristics of a family of functions lets us sketch any one of them without calculating a table of values.

The function $x^2 - 2x + 6$ analysed above is one. Completing the square on the right-hand side of the general form gives

Whatever value $x$ takes, $\dfrac{4ac - b^2}{4a}$ is constant, equal to $K$ say, and $\left(x + \dfrac{b}{2a}\right)^2 \geqslant 0$ as it is a squared quantity. So the function has the general form

1. If $a$ is positive, $f(x)$ is at least equal to $K$: it has a least value of $\dfrac{4ac - b^2}{4a}$, occurring when $x = -\dfrac{b}{2a}$.
2. If $a$ is negative, $f(x)$ can never be greater than $K$: it has a greatest value of $\dfrac{4ac - b^2}{4a}$, occurring when $x = -\dfrac{b}{2a}$.

So $x = -\dfrac{b}{2a}$ is the value of $x$ corresponding to the greatest or least value of $f(x)$. Taking two values of $x$ symmetrical about it, $x = -\dfrac{b}{2a} \pm k$,

that is, input values of $x$ symmetrical about $x = -\dfrac{b}{2a}$ give as output the same value of $f(x)$. From this analysis we deduce that the curve representing $f(x) = ax^2 + bx + c$ is symmetrical about the line $x = -\dfrac{b}{2a}$, called the axis of the curve, and that it turns at a least value when $a > 0$ and at a greatest value when $a < 0$.

The curve representing any particular quadratic function can now be sketched from this information.

A quicker sketch is available when the function factorises. Here

The coefficient of $x^2$ is positive, so $f(x)$ has a least value. When $f(x) = 0$ the corresponding values of $x$ are the roots of $(2x + 1)(x - 4) = 0$, namely $x = -\tfrac12$ and $x = 4$, and the average of these, $\tfrac74$, gives the value of $x$ about which the curve is symmetrical.

Inequalities

Consider the real numbers $5$ and $2$, for which $5 > 2$. The introduction of an extra term on both sides leaves the inequality sign unchanged:

Multiplication of both sides by a positive number also leaves the sign unchanged: $5 \times 2 > 2 \times 2$, that is $10 > 4$. However, if we multiply both sides by a negative number the inequality is no longer true. Multiplying by $-2$, the left-hand side becomes $-10$ and the right-hand side becomes $-4$, and $-10 < -4$. This illustrates the general fact that multiplication or division of both sides by a negative number reverses the inequality sign.

To summarise, if $a$ and $b$ are real numbers such that $a > b$, then

1. $a + k > b + k$ for all real values of $k$;
2. $ak > bk$ for positive values of $k$;
3. $ak < bk$ for negative values of $k$.

These are the rules for manipulating an inequality.

Quadratic Inequalities

An example is $(x - 2)(2x + 1) > 0$. The range of values of $x$ satisfying it can be found graphically. Let $f(x) = (x - 2)(2x + 1)$; the inequality asks for the values of $x$ at which the curve $y = f(x)$ lies above the $x$-axis.

From the sketch we see that $f(x) > 0$, where the curve is above the $x$-axis, for values of $x$ greater than $2$ and for values less than $-\tfrac12$. Therefore the ranges of values of $x$ that satisfy $(x - 2)(2x + 1) > 0$ are

Problems Involving Quadratic Inequalities

Some Other Simple Functions

A table of values for each of the following functions gives enough information to deduce the shape of the curve that represents it.

Exponential Functions

Consider the function $f(x) = 2^x$, for which the following table shows corresponding values of $x$ and $f(x)$.

From this table we see that

1. $f(x) > 0$ for all real values of $x$;
2. as $x$ increases, $f(x)$ increases at a rapidly accelerating rate;
3. $f(x) = 1$ when $x = 0$;
4. as $x$ decreases, through $x = -10, -100, \ldots$, $f(x)$ quickly becomes numerically smaller, and we say that as $x$ approaches minus infinity, $f(x)$ approaches the value zero. This is written $x \to -\infty$, $f(x) \to 0$.

From these observations the sketch of $f(x) = 2^x$ is drawn.

The curve approaches the $x$-axis but never actually touches it or crosses it.

So the $x$-axis is an asymptote to the curve $y = 2^x$. Another way of expressing the behaviour of $f(x)$ for negative values of $x$ is that $f(x)$ approaches a limiting value, or limit, of zero as $x$ approaches minus infinity. This property is written

Any function of the form $a^x$, where $a > 1$, is represented by a curve similar to that deduced for $2^x$.

Rational Functions

For example $\dfrac{1}{x}$, $\dfrac{x}{x - 2}$ and $\dfrac{x^2 - 7}{x + 1}$ are rational functions of $x$. Consider the function $f(x) = \dfrac{1}{x}$ and the following table of corresponding values.

From this table we see that

1. for $x > 0$, $f(x) > 0$, and as $x \to \infty$, $f(x) \to 0$;
2. for $x < 0$, $f(x) < 0$, and as $x \to -\infty$, $f(x) \to 0$, so that $x$ and $f(x)$ always have the same sign;
3. for $x = 0$, $f(0) = \dfrac10$, which has no finite value and is said to be undefined.

In such circumstances we investigate the behaviour of $f(x)$ as $x$ approaches zero. Now $x$ can approach zero in two ways: it can decrease from positive values towards zero, which is to approach zero from above, or increase from negative values towards zero, which is to approach zero from below.

From these tables we see that as $x$ decreases to zero, $f(x) \to \infty$, and as $x$ increases to zero, $f(x) \to -\infty$. From these observations we can draw the sketch representing $f(x) = \dfrac{1}{x}$.

This curve has two asymptotes, the horizontal and the vertical axes. All the curves looked at so far have been unbroken, or continuous, but this curve has a break, or discontinuity, at the point where $x = 0$ and $f(0)$ is undefined. In general, if $f(x)$ is undefined for a finite value of $x$, $x = a$ say, then the curve representing $f(x)$ has a discontinuity where $x = a$.

The function $\dfrac{1}{x}$ does not approach a unique value as $x$ approaches zero: it goes to $\infty$ or to $-\infty$ depending on whether $x$ approaches zero from above or from below. In this case we say that $\displaystyle\lim_{x \to 0} \frac{1}{x}$ does not exist. In general, for $\displaystyle\lim_{x \to a} f(x)$ to exist with a value $k$, say, $f(x)$ must approach $k$ both as $x$ approaches $a$ from above and as $x$ approaches $a$ from below.

Logarithmic Functions

For values of $a > 0$ with $a \neq 1$, any function of the form $\log_a x$, $\log_a (2x + 1)$, and so on, is a logarithmic function, the logarithm being that of Definition 2.21. Consider the function $f(x) = \log_2 x$ and the following table of corresponding values.

From this table we see that

1. when $x = -1$, $f(-1) = \log_2(-1) = b$ say; but there is no real value of $b$ for which $2^b = -1$, since every power of $2$ is positive. So $f(-1)$ does not exist, and all negative values of $x$ lead to the same conclusion: $\log_2 x$ does not exist for negative values of $x$;
2. for $x > 1$, $f(x) > 0$, and as $x \to \infty$, $f(x) \to \infty$;
3. for $x = 0$, $f(0)$ is undefined, so we investigate the behaviour of $f(x)$ as $x$ approaches zero from above. As $x$ decreases to zero, $f(x) \to -\infty$, and for $0 < x < 1$, $f(x) < 0$.

From these observations we can sketch the graphical representation of $f(x) = \log_2 x$.

Any function of the form $\log_a x$ with $a > 1$ has a graph of similar shape. Note that, while $\log_a x$ does not exist for negative values of $x$, the value of $\log_a x$ can itself be negative.

Simple Variations of Functions

The curve representing $f(x) = 2^x$ is now familiar, and from it we can obtain the graphs of some simple variations.

1. $g(x) = 2^x + 1$. Comparing $g(x) = 2^x + 1$ with $f(x) = 2^x$, for any one value of $x$, $g(x)$ is one unit greater than $f(x)$. So the curve representing $g(x)$ is the same shape as that representing $f(x)$ but raised vertically by one unit. In general, the curve representing $f(x) + c$ is the curve representing $f(x)$ raised vertically by $c$ units.
2. $g(x) = 2^{-x}$. For $g(x)$ an input $x = a$ gives the output $2^{-a}$, while for $f(x)$ the input $x = -a$ gives the same output. So $g(a) = f(-a)$, that is $g(x) = f(-x)$ for $x \in \RR$, and the curve representing $g(x)$ is the same as that representing $f(x)$ with the negative and positive values of $x$ transposed. In general, the curve representing $f(-x)$ is the reflection in the vertical axis of the curve representing $f(x)$.
3. $g(x) = -2^x$. Here $g(x) = -f(x)$, so the curve representing $g(x)$ is the same shape as that for $f(x)$ with the positive and negative values of $f(x)$ transposed. In general, the curve representing $-f(x)$ is the reflection in the horizontal axis of the curve representing $f(x)$.

Inverse Functions

Consider the mapping $f : x \mapsto 2x$, $x \in \{2, 3, 4\}$. Under this function the domain $\{2, 3, 4\}$ maps to the image-set $\{4, 6, 8\}$.

It is possible to reverse this mapping: we can map each member of the image-set $\{4, 6, 8\}$ back to the corresponding member of the domain by halving it, $4 \mapsto 2$, $6 \mapsto 3$, $8 \mapsto 4$. Expressed as an algebraic relationship, if $x \in \{4, 6, 8\}$ then $x \mapsto \tfrac12 x$ maps $4 \mapsto 2$, $6 \mapsto 3$, $8 \mapsto 4$.

This reverse mapping is a one-one mapping, so it is a function in its own right, and it is called the inverse function of $f$. Denoting this inverse function by $f^{-1}$, we see that $f^{-1} : x \mapsto \tfrac12 x$, $x \in \{4, 6, 8\}$, reverses the mapping $f : x \mapsto 2x$, $x \in \{2, 3, 4\}$. In fact $f : x \mapsto 2x$ can be reversed for all real values of $x$, so if $f$ is the function defined by $f : x \mapsto 2x$, $x \in \RR$, then $f^{-1}$ is the function which reverses this mapping, defined by

Now consider the function $f : x \mapsto x^2$, $x \in \RR$. This is a many-one mapping: there are two values of $x$ which map to one value of $f(x)$, both $2$ and $-2$ mapping to $4$, for example. We can reverse this mapping by taking the positive and the negative square root of each member of the image-set, which in algebraic form is the mapping $x \mapsto \pm\sqrt{x}$. However, this is a one-many mapping, so it is not a function, and we say that $f : x \mapsto x^2$, $x \in \RR$, does not have an inverse function.

If, however, we restrict the domain of $f$ to $x \geqslant 0$, redefining the function as $f : x \mapsto x^2$, $x \geqslant 0$, then it becomes a one-one mapping. The reverse mapping $x \mapsto \sqrt{x}$ is also one-one, so the function $f : x \mapsto x^2$, $x \geqslant 0$, does have an inverse, namely

The Graphs of Functions and Their Inverses

The graphs of $f(x) = 2x$ and $f^{-1}(x) = \tfrac12 x$ for $x \in \RR$, and of $f(x) = x^2$ and $f^{-1}(x) = \sqrt{x}$ for $x \geqslant 0$, are sketched below. Observing these, we see that in each case the graph of $f^{-1}(x)$ is the reflection of the graph of $f(x)$ in the line $y = x$.

This is true for the graph of any function $f$ and its inverse $f^{-1}$. The line $y = x$ is the graph of $f : x \mapsto x$, and this function is its own inverse. So if the graph representing a function is known, the graph representing its inverse can be sketched. Even when a function $f$ does not possess an inverse, the curve representing $f$ can still be reflected in the line $y = x$, but in that case the reflected graph does not represent a function.

In the same way, for any base $a > 0$ with $a \neq 1$, the function $\log_a x$ from the positive numbers to the real numbers is the inverse of the function $a^x$ from the real numbers to the positive numbers.

Coordinate Geometry