Review of Functions¶

This is a quick run through of functions.

Functions¶

Definition of Function¶

A function \(f\) is a rule of correspondence that associates with each element \(x\) in one set \(D\), called the domain, a single value of \(f(x)\) from a second set, \(R\). The set of all values obtained is called the range of the function.

Linear Function¶

A function whose graph is described as a line is called a linear function. Algebratically denoted

\[f(x)=mx+b=a_1 x + a_0\]

To expand from the previous definition we recall the following important definition.

Definition of Polynomials¶

A function \(P\) is called a polynomial if

\[ P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{2}x^{2}+a_{1}x+a_{0} \]

where \(n\) is a non-negative integer and the numbers \(a_{0},a_{1},...,a_{n}\) are constants called the coefficients of the polynomial. The domain of any polynomial is \(\mathbb{R}=\left\{ x\,|\,x\in(-\infty,\infty)\right\} \). If the leading coefficient \(a_{n}\ne0\), then the degree of the polynomial is \(n\).

Power Function¶

A function of the form

\[f(x)=x^a\]

where \(a\) is a real number is called a power funciton.

If \(a=1,2,3,...\) we have a polynomial.
If \(a=\frac{1}{n}\) where \(n=1,2,3,...\), then we say

\[f(x)=x^{\frac{1}{n}}=\sqrt[n]{x}\]

is a root function. * If \(n\) is even then the domain of \(f\) is \(\{x|x\in[0,\infty)\}\). * If \(n\) is odd then the domain of \(f\) is \(\{x|x\in(-\infty,\infty)\}\).

If \(a=-1,-2,-3,...\), then we say

\[f(x)=x^a=\frac{1}{x^{-a}}\]

is the reciprocal function. * If \(a=-1\), then we have

$$f(x)=x^{-1}=\frac{1}{x^{-(-1)}}=\frac{1}{x^1}=\frac{1}{x}$$

If \(a=0\), then we say

\[f(x)=x^0=1\]

is the constant function.

Absolute Value Function¶

The absolute value function is

\[\begin{split} |x|=\begin{cases} x & \text{if} x\ge 0\\ -x & \text{if} x<0 \end{cases} = \sqrt[2]{x^2} \end{split}\]

Expoential Function¶

A function of the form \(f(x)a^x\), where \(a>0\) and \(a\ne 1\), is called an exponential function with base \(a\). The domain of the exponential function is \(\{ x | x\in (-\infty,\infty) \}\) and the range is \(\{ y | y\in (0,\infty) \}\).

plot2d([exp(x),exp(-x)],[x,-5,5],[y,-1,5],[legend,"f(x)=e^x","f(x)=e^{-x}"]);

svg

Logarithmic Function¶

A function of the form \(f(x)=\log_{a}(x)\), where \(a\ne1\) and \(a>0\), is called a logarithmic function “log base a of x”. The domain of the logarithmic function is \(\{ x | x\in (0,\infty) \}\) and the range of \(\{ y | y\in(-\infty,\infty) \}\).

\[y=a^x \iff x=\log_a(y)\]

plot2d([log(x),log(-x)],[x,-5,5],[y,-5,5],[legend,"y=ln(x)","y=ln(-x)"]);

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Business Calculus

Review of Functions

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