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# Introduction to functions

## Definition of a function

A function is an expression of the form $$\Tred{y} = 2\Tblue{x}+5.$$ It relates an input $\Tblue{x}$ to an output $\Tred{y}$. We sometimes write $y$ as $\Tred{f}(x)$ to show the dependency with $x$. Formally, the function is $\Tred{f}$.

The value of the function $\Tred{y} = 2\Tblue{x}+5$ for the input $\Tgreen{4}$ is $$\Tred{y} = 2\times \Tgreen{4} + 5 = 13.$$

The input can be any number. It is represented by a letter, called the variable. The letter is generally $\Tblue{x}$, but it can be anything.

The functions $\Tred{y}= \Tblue{t}^2$ and $\Tred{y} = \Tblue{x}^2$ are the same, because the letters for the variable and for the function don't matter.

The function $\Tred{y} = 2 \Tblue{x} -3$ can be represented by a table for several inputs.

 $$x$$ $$y$$ $$-2$$ $$-1$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$-7$$ $$-5$$ $$-3$$ $$-1$$ $$1$$ $$3$$ $$5$$

## Introduction to graphs

A simple way to represent a function is by drawing its graph. For each input $\Tblue{x}$ on the horizontal axis, called the $x$-axis, we plot the corresponding output $\Torange{y}$ are on the vertical axis, called the $y$-axis.

The graph of a function gives its value for each possible input. Here, the value for $\Tblue{2}$ is $\Torange{4}$

The graph is a two-dimensional representation of the input / output table associated with a function.

 x y -2 -1 0 1 2 3 4 -7 -5 -3 -1 1 3 5
The function $\Torange{y} = 2 \Tblue{x} -3$ and its graph

## Constant functions

A function is constant if it only ever takes one value. Any input is mapped to the same number by the function. Constant functions are of the form $$y = \Tblue{a}$$ where $\Tblue{a}$ is a fixed number.

The function $y=\Tblue{-2}$ and the function $z = \Tblue{0}$ are both constant functions.

The function $y = 2 x$ is not constant because it takes different values. For example, $y=0$ for $x = 0$ and $y=2$ for $x=1$.

The graph of a constant function is a horizontal straight line.

Graph of the constant function $y = \Tblue{2}$

## Linear functions

A linear function has a straight line as its graph. Linear functions take the form $$y = \Tgreen{a} x + \Tblue{b}$$ for two numbers $\Tgreen{a}$ and $\Tblue{b}$.

All constant functions are linear ($\Tgreen{a}=0$).

$y = \Tgreen{-2}x +\Tblue{2}$ is not constant, but it is linear.

Functions such as $y = x^2$ are not linear.

The number $\Tblue{b}$ is the value of the function at $x=0$. It is the intercept with the $y$-axis.

The number $\Tgreen{a}$ is the gradient of the function, or slope. For any two arbitrary values $x_1\ne x_2$, the gradient is the ratio $$\Tgreen{a} = \frac{y_2 - y_1}{x_2 - x_1}.$$

For $y = -x +2$, we can check that $$\Tblue{b} = 2,\qquad \Tgreen{a} = \frac{1-2}{1} = -1.$$

Graph of the linear function $y = \Tgreen{2}x \Tblue{-1}$. The intercept with the $y$-axis is $\Tblue{-1}$ and the gradient is $\Tgreen{2}$

A quadratic function takes the form $$y = \Tred{a}x^2 + \Tgreen{b}x + \Tblue{c}$$ for three numbers $\Tred{a}\ne0$, $\Tgreen{b}$ and $\Tblue{c}$.

$y = \Tred{-2} x^2 + \Tblue{1}$ and $y = \Tred{2}x^2 \Tgreen{-3} x - \Tblue{6}$ are quadratic.

Constant functions and linear functions are NOT quadratic because they correspond to $\Tred{a}=0$.

The functions $x^3$ and $1/x$ are not quadratic.

• If $\Tred{a}\gt 0$, the graph has a U shape. It is symmetrical across the vertical line through its minimum point.
• If $\Tred{a}\lt 0$, the graph has an inverted U shape. It is symmetrical across the vertical line through its maximum point.

The number $\Tblue{c}$ is the value of the function for $x=0$. It is the intercept with the $y$-axis.

Graphs of the two quadratic functions $y = 2x^2-2x-2$ (left) and $y = - x^2 +4$ (right) and a linear function $y = 2x + 2$ (centre).

## Reciprocal functions

A reciprocal function takes the form $$y = \frac{\Tblue{a}}{x}$$ for a number $\Tblue{a}$. We can also write the function as $\Tblue{a}/x$ or $\Tblue{a} x^{-1}$.

$\displaystyle\frac{\Tblue{1}}{x}$ and $\displaystyle\frac{\Tblue{-2}}{x}$ are reciprocal functions.

The number $\Tblue{a}$ is the value of the function at $x=1$.

The reciprocal function is defined for $x\ne0$. It has no value at $0$.

For positive $x$, $y = \Tblue{1}/x$ becomes bigger as $x$ gets smaller. $$f(1) = 1,\; f(0.5) = 2,\; f(0.1) = 10,\; f(0.01) = 100.$$ As positive values of $x$ tend to $0$, the graph gets closer to the upper $y$-axis but never crosses it. Similarly, for negative $x$, the graph gets closer to the lower $y$-axis.

We say that the $y$-axis is a vertical asymptote of $1/x$.

Graphs of the reciprocal functions $y= 1/x$ (left), $y = 3/x$ (centre) and $y = - 2/x$ (right).

## Power function

A power function is a function of the form $$y = \Tblue{a} x^{\Torange{n}}.$$ The number $n$ is the power. It is often an integer.

Power functions have special name depending on the value of the power.

 function power $\Torange{n}$ $2^{\Torange{n}}$ reciprocal constant linear quadratic cubic $-1$ $0$ $1$ $2$ $3$ $0.5$ $1$ $2$ $4$ $8$
Graph of the power functions $x^n$ for several values of $n$

A power function with a negative exponent is not defined for $x=0$.

A number raised to a negative power is the reciprocal of the number at the positive power. $$\Tblue{3}^{-\Torange{2}} = \frac{1}{\Tblue{3}^{\Torange{2}}} = \frac{1}{\Tgreen{9}}.$$

The graph of the sum of power functions is obtained by adding, for each $x$, the $y$-values of each individual function.

Graph of the sum of power function $y=x^{-2} - 2x.$ The graph of $y=x^{-2}$ is on the left, $y=-2x$ in the centre, and the sum on the right.

## Exponential function

An exponential function is a function of the form $$y = \Tblue{k} \Torange{a}^{x}.$$ The number $\Torange{a}$ is positive.

The computation without a calculator is difficult for a general $x$. However, it can be done when $x$ is rational.

• When $x$ is an integer: $y$ is the usual $\Torange{a}$ to the power of $x$. $$2^{\Tred{1}} = 2,\; 2^{\Tred{2}} = 2\times2 = 4,\; 2^{\Tred{3}} = 2^2\times 2 = 8,\; 2^{\Tred{4}} = 16.$$
• When $x$ is the reciprocal of an integer: $y$ is the $x$-th root of $\Torange{a}$, i.e. the number that is $\Torange{a}$ when raised to the power $x$ $$2^{1/\Tred{2}} = \sqrt{2} \simeq 1.41,\; 2^{1/\Tred{3}} = \sqrt[\Tred{3}]{2}\simeq 1.26,\; 2^{1/\Tred{4}} = \sqrt[\Tred{4}]{2}\simeq 1.19.$$
• When $x$ is rational ($p/q$): $y$ is the $q$-th root of $\Torange{a}$ raised to the power $p$ $$2^{\Tblue{3}/\Tred{2}} = (\sqrt{2})^\Tblue{3} \simeq (1.41)^\Tblue{3}\simeq 1.58,\; 2^{\Tblue{2}/\Tred{3}} = (\sqrt[\Tred{3}]{2})^\Tblue{2}\simeq 2.83$$
Graph of the exponential function $y = 2^x$