Domain and range of a function

If $y = f(x)$, that is, $y$ is a function of $x$, then the domain is the set of all $x$-values that satisfy the equation $y = f(x)$.

If $y = f(x)$, that is, $y$ is a function of $x$, then the range is the set of all $y$-values satisfying the equation $y = f(x)$.

Example 1

Let $f(x) = 3x - 5$ for all values of $x$ such that $-2 \le x \le 3$. Find its range.

Solution:

$f(x) = y = 3x - 5$

When $x = -2$:

$f(-2) = y = 3(-2) - 5 = -11$, so $(x, y) = (-2, -11)$

When $x = 3$:

$f(3) = y = 3(3) - 5 = 4$, so when $x = 3$, $y = 4$

Therefore, $y$ is found between $-11$ and $4$.

Range:

$\{ y: -11 \le y \le 4 \}$

Example 2

If $f(x) = x^2 - 3$, state the domain and range of $f(x)$.

Solution:

Domain: All real numbers.

Range:

$f(x) = y = x^2 - 3$

Make $x$ the subject:

$y + 3 = x^2$

This is a quadratic function, and since the square of any real number is non-negative, the minimum value of $y$ is $-3$. Thus, the range is $y \ge -3$.

Range:

$\{ y: y \ge -3 \}$

One-to-One Functions:

A one-to-one function is a function in which each element from the domain is mapped to exactly one element in the range. In other words, if $a \ne b$, then $f(a) \ne f(b)$.

Examples of one-to-one functions:

• $f(x) = 3x + 2$
• $f(x) = x + 6$
• $f(x) = x^3 + 1$

Many-to-One Functions:

This is a function where two or more elements from the domain can have the same image (i.e., they map to the same value in the range).

Examples of many-to-one functions:

• $f(x) = x^2 + 1$
• $f(x) = x^4 - 2$

Note:

• All functions with odd degrees (such as $x^3 + 1$) are one-to-one functions.
• All functions with even degrees (such as $x^2 + 1$) are many-to-one functions.

Example 3

Let $A = \{-2, -1, 0, 1, 2\}$ and $B = \{0, 1, 4\}$ and the function $f$ mapping each element from set $A$ to those of $B$ is defined as $f(x) = x^2$. Is $f$ one to one function?

i.e $f(a) = f(b)$ but $a \ne b$

Example 4

Let $P = \{-2, -1, 0, 1, 2\}$ and

$Q = \{-1, 0, 1, 2, 3\}$

$g(x) = x + 1$, is $g$ one to one function?

Solution:

$g(x)$ is one to one function because every element in $P$ has only one image in $Q$.

NB: In example 1, $f(x)$ is not a one to one function because $-2$ and $2$ in $A$ have the same image in $B$, that is $4$ is the image of both $2$ and $-2$. Also $1$ is the image of both $1$ and $-1$.

Example 5

State whether or not if the following graphs represent a one to one function:

Solution:

Draw a line parallel to the $x$ axis and see if it crosses the graph at more than one points. If it does, then, the function is many to one and if it crosses at only one point then the graph represents a one to one function.