Chapter 2, Problem 1CC

To determine

(a)

To define:

The function.

Explanation of Solution

A function is a relation between a set A to set B that relates each element of set A to unique element of set B.

As an example consider a function defined for set of real numbers to set of real numbers as $f\left(x\right)=2x+3$.

$\begin{array}{l}f:R\to R\\ f(x)=2\cdot x+3\end{array}$

Here, the function is defined from real number to real number and variable in the function is x.

To determine

(b)

To define:

The domain and range of a function.

Explanation of Solution

A functionf, which relates each element of set A to unique element of set B.

The set of each element of set A which is an input for function f is called the domain of function f.

The set of each element of set B which is an output of function f is called the range of a function f.

Also, the domain of a function $f\left(x\right)$ is the range of its inverse function $f^{-1}\left(x\right)$ and vice-versa.

To determine

(c)

To define:

The graph of a function.

Explanation of Solution

Assume a functionfis a relation between a set A to set B that relates each element of set A to unique element of set B.

The graphof a functionfis the set of allordered pairs $\left(x,f\left(x\right)\right)$, such thatxis in thedomain of the functionf, wherexand $f\left(x\right)$ arereal numbers.

These pairs are Cartesian coordinates of points in the Euclidean plane.

To determine

(d)

To define:

The dependent and independent variable.

Explanation of Solution

A symbol that stands for an arbitrary input is called anindependent variable, while a symbol that stands for an arbitrary output is called adependent variable.

In a function f, the domain is the input for function which is independent variable as it is taken by choice.

The range is output in function f which depends upon the choice of input, so it is a dependent variable.

For example, consider a function $y=2x+3x^2$. Here, the variable x is a dependent variable and the variable y is an independent variable.