Functions & Graphs | Complete Guide with Concepts & Examples

Functions are a fundamental concept in mathematics, helping us understand the relationships between variables and how changes in one variable influence another. This article delves into key aspects of functions, from definitions to properties, examples, and solved problems, all presented in an engaging and easy-to-follow manner.

What is a Function?

A function establishes a relationship between two quantities. For example, the area of a square changes based on the length of its side. This relationship can be expressed mathematically. Let the variable $xx$ take values from the set $DD$ , the domain of the function. A function is a rule that assigns every value of $xx$ in $DD$ a unique value of $yy$ , called the dependent variable.

Key Terms:

Independent Variable: $xx$
Dependent Variable: $yy$
Domain: Set $DD$ containing all possible $xx$ values.
Range: Set of all possible $yy$ values.

This is symbolized as $y=f(x)y = f(x)$ , where $ff$ denotes the rule governing the relationship.

Even and Odd Functions

Even Functions

A function $f(x)f(x)$ is even if $f(x)=f(-x)f(x) = f(-x)$ for all $xx$ in its domain.

Properties of Even Functions:

The sum, difference, product, and quotient of even functions are also even.
Their graphs are symmetrical about the $yy$ -axis.

Examples:

$y=x2y = x^2$
$y=x4y = x^4$
$y=∣x∣y = |x|$
$y=cos⁡(θ)y = \cos(\theta)$

Odd Functions

A function $f(x)f(x)$ is odd if $f(-x)=-f(x)f(-x) = -f(x)$ for all $xx$ in its domain.

Properties of Odd Functions:

The sum and difference of odd functions are also odd.
The product and quotient of odd functions are even.
Their graphs are symmetrical about the origin.

Examples:

$y=xy = x$
$y=x3y = x^3$
$y=sin⁡(x)y = \sin(x)$

Tip: Not all functions are strictly even or odd. However, any function can be expressed as the sum of an even function and an odd function.

The Inverse of a Function

An inverse function reverses the roles of the independent and dependent variables. For $y=f(x)y = f(x)$ with domain $DD$ and range $RR$ , its inverse $x=g(y)x = g(y)$ swaps $DD$ and $RR$ :

Domain of $ff$ : $DD$
Range of $ff$ : $RR$

Graphs of inverse functions are symmetrical about the line $y=xy = x$ .

Example:

Given $y=4x2y = 4x^2$ :

Swap $xx$ and $yy$ : $x=4y2x = 4y^2$ .
Solve for $yy$ : $y=±x/2y = \pm\sqrt{x}/2$ .

Plotting Functions

To visualize a function $y=f(x)y = f(x)$ , plot $xx$ -values on the horizontal axis and corresponding $f(x)f(x)$ values on the vertical axis. For example, consider a table of $xx$ and $f(x)f(x)$ values to draw the graph.

Shifting Graphs

Understanding how graphs shift when the function changes is crucial:

Vertical Shifts: y=f(x)+cy = f(x) + c
- Shifts the graph $cc$ units up (if $c>0c > 0$ ) or down (if $c<0c < 0$ ).
Horizontal Shifts: y=f(x±c)y = f(x \pm c)
- Shifts the graph $cc$ units left (if $+c+c$ ) or right (if $-c-c$ ).

Solved Examples

1. Domain of $f(x)=1x+2f(x) = \frac{1}{x+2}$ :

The function is undefined when $x+2=0x + 2 = 0$ , or $x=-2x = -2$ .
Domain: All real numbers except $-2-2$ .

2. Range of $y=x2y = x^2$ :

$y≥0y \geq 0$ , as squaring any real number yields a non-negative result.
Range: $y≥0y \geq 0$ .

3. Domain and Range of Sine Function:

Domain: All real numbers.
Range: $−1≤sin⁡(x)≤1-1 \leq \sin(x) \leq 1$ .

4. Inverse of $y=4x2y = 4x^2$ :

Swap $xx$ and $yy$ : $x=4y2x = 4y^2$ .
Solve for $yy$ : $y=±x/2y = \pm\sqrt{x}/2$ .

Conclusion

Functions are vital for understanding mathematical relationships. From even and odd functions to inverses and graphical transformations, mastering these concepts equips you with problem-solving skills across various fields.

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