
# 1.0: Library of functions

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Thinking out Loud

What is the meaning of a real-valued function?

### Real Valued functions

Definition

Let $$D \subset \mathbb{R}$$. Then $$f:D\to \mathbb{R}$$ is said to be a real valued function from $$D$$ to  $$\mathbb{R}$$ if for each element $$x \in D$$, there exists a unique (one and only one) element  $$y \in \mathbb{R}$$, such that $$y=f(x).$$ In this case $$D$$ is called domain of $$f$$, $$x$$ is called an independent variable, and $$y$$ is called an dependent variable,

Note that we will follow the following convention: for a given real valued function $$f$$ , the domain of f is denoted by $$D(f)$$ and defined by $$D(f) =$$ the set of all $$x$$ such that  there exists a $$y \in \mathbb{R}$$ such that  $$y=f(x)$$.

## Library of functions

The following examples  of functions will be used in this course:

### Polynomial functions

Definition

#### Linear functions

Example $$\PageIndex{1}$$

Define $$f: \mathbb{R}\to \mathbb{R}$$ by $$f(x)=x+1$$. Then  $$D(f) = \mathbb{R}$$.

Exercise $$\PageIndex{1}$$

Define $$f: \mathbb{R}\to \mathbb{R}$$ by $$f(x)=x-1$$. Find the domain.

$$D(f) = \mathbb{R}$$

$$\rm I\!R$$

Example $$\PageIndex{2}$$:

Define $$f: \mathbb{R}\to \mathbb{R}$$ by $$f(x)=x^2$$. Then  $$D(f) = \mathbb{R}$$.

Exercise $$\PageIndex{2}$$

Define $$f: \mathbb{R}\to \mathbb{R}$$ by $$f(x)=-x^2$$. Find the domain.

$$D(f) = \mathbb{R}$$

#### Cubic functions

Example $$\PageIndex{3}$$:

Define $$f: \mathbb{R}\to \mathbb{R}$$ by $$f(x)=x^3$$. Then  $$D(f) = \mathbb{R}$$.

.

Exercise $$\PageIndex{3}$$

Define $$f: \mathbb{R}\to \mathbb{R}$$ by $$f(x)=x^3$$. Find the Domain

$$D(f) = \mathbb{R}$$

### Rational functions

Example $$\PageIndex{4}$$

Define $$f: \mathbb{R}\to \mathbb{R}$$ by $$f(x)= \displaystyle\frac{x^2-1}{x-1}$$. Then  $$D(f)= \mathbb{R} \setminus \{1\}$$.

Exercise $$\PageIndex{4}$$

Define $$f: \mathbb{R}\to \mathbb{R}$$ by $$f(x)= \displaystyle\frac{x^2-1}{x+1}$$. Find the domain.

$$D(f)= \mathbb{R} \setminus \{-1\}$$

Example $$\PageIndex{5}$$:

Define $$f: \mathbb{R}\to \mathbb{R}$$ by $$f(x)= \displaystyle\frac{1}{x-1}$$. Then  $$D(f)= \mathbb{R} \setminus \{1\}$$.

Exercise $$\PageIndex{5}$$

Define $$f: \mathbb{R}\to \mathbb{R}$$ by $$f(x)= \displaystyle\frac{1}{x-1}$$. Find the domain.

$$D(f)= \mathbb{R} \setminus \{-1\}$$.

### Irrational functions

upper half unit circle $$\PageIndex{6}$$

Define $$f: \mathbb{R}\to \mathbb{R}$$ by $$f(x)=\sqrt{1-x^2}$$.  Then  $$D(f)= [-1,1].$$

Exercise $$\PageIndex{6}$$

Define $$f: \mathbb{R}\to \mathbb{R}$$ by $$f(x)=\sqrt{4-x^2}$$.  Find the domain.

$$D(f)= [-2,2].$$

Example $$\PageIndex{7}$$:

Define $$f: \mathbb{R}\to \mathbb{R}$$ by $$f(x)=\sqrt{1-x}$$.  Then $$D(f)= [0,\infty)$$.

Exercise $$\PageIndex{7}$$

Define $$f: \mathbb{R}\to \mathbb{R}$$ by $$f(x)=\sqrt{1-x}$$. Find the domain.

$$D(f)= [0,\infty)$$

### Piecewise-defined function

#### Absolute valued function

Example $$\PageIndex{8}$$

Define $$f: \mathbb{R}\to \mathbb{R}$$ by $$f(x)=|x|$$.

Note that $$\sqrt{x^2}=|x|$$.

Exercise $$\PageIndex{1}$$

Define $$f: \mathbb{R}\to \mathbb{R}$$ by $$f(x)=|x|+1$$

#### Step functions

Example $$\PageIndex{9}$$: Sign function

Define $$f: \mathbb{R\setminus\{0\}}\to \mathbb{R}$$ by $$f(x)=\frac{|x|}{x}$$.

Example $$\PageIndex{10}$$: Unit Step function

Define $$f: \mathbb{R}\to \mathbb{R}$$ by $$f(x)=$$.