Functions

MATH1051

2.1 Definition: Function, Domain, Range

Let $X$ and $Y$ be subsets of $\mathbb{R}$. A function $f:X\to Y$ is a rule which assigns to every element $x\in X$ exactly one element $f(x)\in Y$ called the value of $f$ at $x$. Here $X$ is called the domain of $f$ and

$$f(X)=\{f(x)\mid x\in X\}$$

is called the range of $f$, also written range$(f)$.

The range of $f$, $f(X)$, is a subset of $Y$. The range is the set of all possible values of $f(x)$ as $x$ varies throughout the domain. Note that $f(X)$ is not necessarily equal to all of $Y$.

2.2 Graphs

We can represent a function by drawing its graphs which is the set of all points $(x,y)$ in a plane where $y=f(x)$.

2.3 Convention (Domain)

An expression like “the function $y=\sqrt{1-x^2}$” means the function $f$ with $y=f(x)=\sqrt{1-x^2}$. When the domain is not specified it is taken to be the largest subset of $\mathbb{R}$ on which the rule is defined (and gives a real output). In this example, the domain would be $[-1,1]$.

2.4 Vertical line Test

Not every curve represents the graph of a function. The crucial function property states that for each value $x$ in the domain there must correspond exactly one value $y$ in the range. Thus in the graph of a function, any vertical line $x=$ constant must cut the graph in at most one point.

The graph for the circle $x^2+y^2=1$, we can say that the vertical line intersects the circle at two points. In this case, the two $y$ values are given by $y=\pm \sqrt{1-x^2}$. Therefore $x^2+y^2=1$ does not give rise to a function on any domain intersecting $(-1,1)$.

2.5 Exponential Functions

An exponential function is one of the form $f(x)=a^x$, where the base $a$ us a positive constant, and $x$ is said to be the exponent or power. One very common exponential function which we shall see often in this course is given by $f(x)=e^x$. It cuts the $y-$axis at $y=1.$

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f(x)=2^x

Exponential functions are very useful for modelling many natural phenomena such as population growth (base $a>1$) and radioactive decay (base $0<a<1$).

2.7 One-to-one Functions

A function $f:X\to Y$ is said to be one-to-one or injective if $\forall x_1,x_2\in X$.

On the graph of $f$, the 1-1 property holds exactly if any horizontal line $y=$ constant cuts through the curve in at most one place.

If one $x$ corresponds to another $x$, then it is not one-to-one. Similarly, if $y$ corresponds to another $y$ of the same value, then the function is not one-to-one.

2.7.1 Example

If the value of the function $y=x^2+3$:

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y=x^2+3

Then this is not a one-to-one function. This is because it does pass the vertical line test (with one $x$ corresponding to one $x$ only), but it does not pass the horizontal line test (for example, $x=2\cap x=-2$ yield the same $y$ value (with $y=7$)).

On the other hand, for example, $y=e^{2x}$:

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y=exp(2x)

2.8 Inverse Functions

Let $f:X\to Y$ be a 1-1 function. For each $y\in f(X)$ the range of $f$, there is a unique $x$ with $f(x)=y$. Define the inverse function $f^{-1}:f(X)\to X$ by $f^{-1}(y)=$ that unique $x\in X$ with $f(x)=y$. So $f^{-1}(y)=x⟺y=f(x).$

The inverse function reverses the direction of the mapping.

2.9 How to Find $f^{-1}$

$f(x)=y⟹f^{-1}(y)=x$ $\therefore$ To find $f^{-1}$ solve for $x$ in terms of $y$.

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f(x)=(2x-5)^(1/3)

2.10 Logarithms

Logarithms are the inverse functions of the exponential functions.

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y=2^x
y=3^x
y=4^x

Three plots of $y=a^x$. (Green = $4^x$) (Red = $3^x$) (Blue = $2^x$)

From the graph of $y=a^x$ ($a\ne 1$ a positive constant), we see that it is 1-1 and thus has an inverse, denoted ${\log }_{a}x$. From this definition we have the following facts:

$$\begin{array}{rlr}{\log }_a(a^x)& =x& \forall x\in \mathbb{R},\\ a^{{\log }_{a}x}& =x& \forall x>0.\end{array}$$

2.11 Natural Logarithm

Now we set $a=e$ (Euler’s number $=2.71828\dots$) The inverse function of $f(x)=e^x$ is:

$${\log }_e(x)\equiv \ln x$$

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y=exp(x)
y=log(x)

2.12 Inverse Trigonometric Functions

The function $y=\sin x$ is 1-1 if we just define it over the interval $\left[\frac{\pi }{2},\frac{\pi }{2}\right]$. The inverse function for this part of $\sin x$ is denoted $\arcsin x$. Thus $\arcsin x$ is defined on the interval $[-1,1]$ and takes values in the range $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$. The graph can easily be obtained by reflecting the graph of $\sin x$ about the line $y=x$ over the appropriate interval.

Similarly $y=\cos x$ is 1-1 on the interval $[0,\pi ]$ and its inverse function is denoted $\arccos x$. The function $\arccos x$ is defined on $[-1,1]$ and takes values in the range $[0,\pi ]$.

Also, $\tan$ is 1-1 on the open interval $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ with inverse function denoted by $\arctan x$. Hence $\arctan$ has the domain $(-\infty ,\infty )$ with values in the range $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$.

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f^(-1)(x)=(exp(x)/(1+2*exp(x)))