Domain, Codomain, and Range of Functions

The domain, codomain, and range are fundamental concepts in understanding functions. The domain refers to the set of all possible input values for which the function is defined. The codomain is the set of all potential outputs, or the set where the function's values could lie. The range is the actual set of outputs that the function produces based on the inputs from the domain. Understanding these concepts helps in analyzing how a function behaves and the relationship between its inputs and outputs.

1.0What Is Domain, Codomain, and Range?

• Domain of a Function: The domain of a function is the set of all possible input values (usually represented by x) for which the function is defined. In other words, it’s the set of values you can plug into the function to get a valid output. For example, in the function $f(x)=\frac{1}{x}$, the domain is all real numbers except x = 0 because division by zero is undefined.
• Codomain of a Function: The codomain is the set of all possible output values (usually represented by y) that a function could potentially produce. The codomain is often specified when defining the function, but it may not necessarily include all the values the function actually produces. The codomain can be thought of as the "target" set, or the set of possible values the function maps to.
• Range of a Function: The range is the set of all actual output values that a function can produce. Unlike the codomain, the range only includes the values that are actually obtained when you input values from the domain into the function. It’s essentially the "realized" output of the function.

2.0Difference Between Domain, Codomain, and Range Functions

While these three terms are related, they have distinct meanings:

1. Domain refers to the possible input values, or the set from which the function draws its inputs.
2. Codomain refers to the set of potential output values, or the set where the function’s outputs are expected to lie.
3. Range refers to the set of values the function actually produces from the domain. It is a subset of the codomain.

Thus, the range can be a smaller subset of the codomain. While the domain is usually defined by the function itself, the codomain is typically given by the person defining the function. The range, however, is determined based on how the function operates and what outputs it actually produces.

Examples of Domain, Codomain, and Range

Example 1: f(x) = x²

• Domain: The domain of this function is all real numbers, i.e., $xϵ\mathbb{R}$, because we can input any real number into this function.
• Codomain: The codomain of this function is often taken as the set of all real numbers, $\mathbb{R}$.
• Range: The range of the function f(x) = x² is y ≥ 0, because the square of any real number is always non-negative. So, the range is $[o,\infty )$.

Example 2: $f(x)=\frac{1}{x}$

• Domain: The domain of this function is all real numbers except x = 0, because division by zero is undefined. So, the domain is $xϵ\mathbb{R},x\ne 0$.
• Codomain: The codomain is typically taken as all real numbers, $\mathbb{R}$.
• Range: The range of this function is also all real numbers except y = 0, because there’s no value of x that gives an output of 0. Hence, the range is $yϵ\mathbb{R},y\ne 0$.

Example 3: f(x) = sin(x)

• Domain: The domain of the sine function is all real numbers, $xϵ\mathbb{R}$, because sine is defined for every real number.
• Codomain: The codomain of the sine function is typically $\mathbb{R}$, the set of real numbers.
• Range: The range of the sine function is between –1 and 1, inclusive. Therefore, the range is [–1, 1].

3.0How to Find Domain, Codomain, and Range of Functions

1. Finding the Domain:

• Identify any values of xx that would make the function undefined (e.g., division by zero, taking the square root of a negative number).
• The domain will consist of all valid values of xx for which the function is defined.

2. Finding the Codomain: The codomain is typically specified when defining the function. It is the set from which the output values are expected to be drawn.
3. Finding the Range:

• The range can be found by analyzing the behavior of the function and determining what actual output values it produces for every possible input in the domain.
• You can sometimes use graphing to help determine the range or use algebraic methods (e.g., solving inequalities or analyzing the function’s behavior).

4.0Solved Questions on Domain, Codomain, and Range of Functions

Example 1: Find the domain, codomain, and range of the function: $f(x)=\sqrt{(}x)$

• Domain: Since we can’t take the square root of a negative number, the domain is x ≥ 0.
• Codomain: The codomain is typically $\mathbb{R}$, the set of real numbers.
• Range: The range is y ≥ 0, as the square root of any non-negative number is always non-negative.

Example 2: Find the domain, codomain, and range of the function: $f(x)=\frac{2}{x-3}$

• Domain: The function is undefined when x = 3, so the domain is $xϵ\mathbb{R},x\ne 3$.
• Codomain: The codomain is $\mathbb{R}$, the set of real numbers.
• Range: The range is $yϵ\mathbb{R},y\ne 0$, because y = 0 is never reached (as the function cannot give a zero output).

Example 3: Find the domain, codomain, and range of the function: f(x) = 2x + 3, where x is a real number.

Solution:

• Domain: The domain of a function is the set of all possible input values for which the function is defined. For f(x) = 2x + 3, the function is defined for all real values of x. Thus, the domain of f(x) is $\mathbb{R}$ (the set of all real numbers).
• Codomain: The codomain is the set of all possible outputs that the function can produce, as specified by the function. In this case, we haven't explicitly mentioned the codomain. But if no restrictions are given, we typically assume that the codomain is also $\mathbb{R}$ because the output is a real number for every input.
• Range: The range is the set of actual output values of the function for all inputs from the domain. Since f(x) = 2x + 3 is a linear function, it can take any real value as its output. Therefore, the range of f(x) is $\mathbb{R}$.

Summary:

• Domain: $\mathbb{R}$
• Codomain: $\mathbb{R}$
• (typically assumed)
• Range: $\mathbb{R}$

Example 4: Let $f(x)=\sqrt{x-2}$, where x is a real number.

Solution:

• Domain: The domain of f(x) consists of the values of x for which the expression inside the square root is non-negative. That is, x – 2 ≥ 0, which gives x ≥ 2. Therefore, the domain of f(x) is [2, ∞).
• Codomain: Again, we assume that the codomain is $\mathbb{R}$, since the output of the function could theoretically be any real number. However, this will be clarified by the range.
• Range: Since $f(x)=\sqrt{x-2}$ represents a square root function, the output must be non-negative. The minimum value occurs when x = 2, and f(x) = 0. As x increases, f(x) increases as well. Therefore, the range of f(x) is [0, ∞).

Summary:

• Domain: [2, ∞)
• Codomain: (assumed)
• Range: [0, ∞)

Example 5: Let $f(x)=\frac{1}{x-1}$, where x is a real number.

Solution:

• Domain: The domain of the function consists of all xx values for which the denominator is not zero. Therefore, x – 1 ≠ 0, which implies x ≠ 1. Hence, the domain of f(x) is $\mathbb{R}$ (all real numbers except 1).
• Codomain: As no explicit restrictions are given, we assume the codomain is $\mathbb{R}$.
• Range: The function $f(x)=\frac{1}{x-1}$ can take any real value except 0. This is because the function approaches 0 but never reaches it. Therefore, the range of f(x) is $\mathbb{R},\text{0}$.

Summary:

• Domain: $\mathbb{R}$, {1}
• Codomain: $\mathbb{R}$ (assumed)
• Range: $\mathbb{R},\text{0}$

Example 6: Let f(x) = sin(x), where x is a real number.

Solution:

• Domain: The sine function is defined for all real numbers, so the domain of f(x) = sin(x) is $\mathbb{R}$.
• Codomain: The sine function produces values between -1 and 1, inclusive. Thus, the codomain of f(x) is [−1, 1].
• Range: The sine function actually takes all values between -1 and 1. Therefore, the range of f(x) = sin(x) is [−1, 1].

Summary:

• Domain: $\mathbb{R}$
• Codomain: [−1, 1]
• Range: [-1, 1]

Example 7: Let f(x) = x², where x is a real number.

Solution:

• Domain: The function ff(x) = x² is defined for all real values of x. Therefore, the domain is $\mathbb{R}$.
• Codomain: The codomain is typically assumed to be $\mathbb{R}$, unless otherwise specified.
• Range: Since f(x) = x² produces only non-negative outputs, the range of f(x) is [0, ∞).

Summary:

• Domain: $\mathbb{R}$
• Codomain: $\mathbb{R}$(assumed)
• Range: [0, ∞)

5.0Practice Questions on Domain, Codomain, and Range of Functions

Question 1: Find the domain, codomain, and range of the function:  $f(x)=\frac{1}{x-2}$

Question 2: Determine the domain, codomain, and range of the function:  $f(x)=\sqrt{x-3}$

Question 3: For the function $f(x)=\frac{x+4}{x^2-4}$, find the domain, codomain, and range.

Question 4: Consider the function f(x) = $\ln (x+1)$. What is its domain, codomain, and range?

Question 5: Find the domain, codomain, and range of the function: f(x) = 2x + 5 where  $xϵ\mathbb{R}$

Question 6: For the function $f(x)=\frac{x}{\sqrt{x+2}}$, find its domain, codomain, and range.

Question 7: Find the domain, codomain, and range of the following quadratic function:$f(x)=x^2-4x+3$

Question 8: For the function f(x) = sin(x), find its domain, codomain, and range.

Question 9: Determine the domain, codomain, and range for the function:  $f(x)=\frac{3x-1}{x+2}$

Question 10:For the piecewise function $f(x)=\{\begin{array}{ll}x+1& \text{for }x\ge 0\\ -x+2& \text{for }x<0\end{array}$, find the domain, codomain, and range.