Domain and Range Worksheet

Domain refers to the set of all possible values of x for which a function is valid. It tells us the all values that can be put into the function without having issues. Whereas, Range is the set of all possible output values of y that a function can give as output. It tells the values that result from putting the domain into the function.

In this article, we will look at different types of questions related to this important concept in mathematics.

What is Domain and Range?

Domain: The set of all possible input values especially x-values for which the function is defined is known as the domain.

Range: The set of all possible output values mostly y-values that the function can produce is known as the range. It represents the set of values the function can take as its output for the values in its domain.

Let's consider some examples for better understanding.

Example 1: The given function f(x) = \frac{1}{x - 2}​ find its domain and range?

Solution:

• Domain: All real numbers except x = 2.
• Range: All real numbers except y = 0.

Example 2: The given function g(x) = \sqrt{x + 3}​ find its domain and range?

Solution:

Domain: x ≥ − 3.

Range: y ≥ 0.

How to Find Domain?

For a polynomial function f(x): The domain is all real numbers.

Rational function \frac{P(x)}{Q(x)}​: The domain is all real numbers except where Q(x) = 0.

Square root function \sqrt{f(x)}​: The domain is the set of all x such that f(x) ≥ 0.

Even root function \sqrt[n]{f(x)} (where n is even): The domain is f(x) ≥ 0.

How to Find Range?

For Linear function f(x) = mx + b: The range is all real numbers R.

For a quadratic function f(x) = ax² + bx + c:

• If a > 0: The range is y ≥ f(vertex).
• If a < 0: The range is y ≤ f(vertex).

Square root function f(x) = \sqrt{g(x)}​: The range is y ≥ 0.

Rational function \frac{P(x)}{Q(x)}​: The range is determined by solving y = \frac{P(x)}{Q(x)}for x and finding where the function is defined and continuous.

Domain and Range: Solved Problems

Problem 1: Find the domain and range of f(x) = \frac{2x + 3}{x - 1}​.

Solution:

Domain: The function is undefined when the denominator is zero. Set x − 1 = 0, so x = 1. The domain is all real numbers except x = 1, or Domain = {x∈R∣x≠1}.

Range: To find the range, solve for x in terms of y:

y = \frac{2x + 3}{x - 1} ⇒y(x−1)=2x+3⇒xy−y=2x+3⇒x(y−2)=y+3⇒x=y+3​. The range is all real numbers except y = 2, so Range = {y∈R∣y≠2}.

Problem 2: Find the domain and range of f(x) = \sqrt{x - 4}​.

Solution:

Domain: The square root is defined when the expression under the root is non-negative. So, x − 4 ≥ 0, which gives x ≥ 4. The domain is = {x∈R∣x ≥ 4}.

Range: Since y = x − 4​, and a square root always yields non-negative values, the range is Range = {y∈R∣y ≥ 0}.

Problem 3: Determine the domain and range of f(x) = x² − 5x + 6.

Solution:

Domain: This is a quadratic function, so it is defined for all real numbers. The domain is Domain = R.

Range: The quadratic opens upwards (since the coefficient of x² is positive). The vertex of the parabola gives the minimum value. Find the vertex by using the formula x = -\frac{b}{2a}​: x = -\frac{-5}{2(1)} = \frac{5}{2} = 2.5 Substituting x = 2.5 into the function to find the minimum value: f(2.5) = (2.5)² − 5(2.5) + 6 = 6.25 − 12.5 + 6 = − 0.25 The range is Range = {y∈R∣y ≥ − 0.25}.

Problem 4: Find the domain and range of f(x) = \frac{1}{x^2 + 1}​.

Solution:

Domain: The denominator x² + 1 is never zero, so the function is defined for all real numbers. The domain is Domain = R.

Range: The minimum value of x² is 0 (when x = 0), so the minimum value of f(x) is 11=1\frac{1}{1}. The function f(x) approaches 0 as x increases or decreases without bound, but never reaches 0. Thus, the range is Range = {y∈R∣0 < y ≤ 1}.

Problem 5: Determine the domain and range of f(x) = ln⁡(x − 2).

Domain: The natural logarithm function is defined when its argument is positive. So, x − 2 > 0 gives x > 2. The domain is Domain = {x∈R∣x > 2}.

Range: The natural logarithm function can produce all real numbers. The range is Range = R.

Problem 6: Find the domain and range of the function f(x) = 2x + 3.

Solution:

Domain: The function is defined for all real numbers.

Domain = ( − ∞,∞)

Range: As x can take any real value, the output can also be any real number.

Range = ( − ∞,∞)

Problem 7: Determine the domain and range of the function f(x) = \sqrt{x - 2}.

Solution:

Domain: The expression inside the square root must be non-negative: x − 2 ≥ 0⇒x ≥ 2x - 2

Domain = [2,∞)

Range: The square root function produces non-negative outputs. Range = [0,∞)

Problem 8: Find the domain and range of f(x) = 1/x − 4​.

Solution:

Domain: The denominator cannot be zero: x − 4 ≠ 0 ⇒ x ≠ 4

Domain = ( − ∞, 4) ∪ (4, ∞)

Range: The function can produce any real number except zero. Range = ( − ∞, 0) ∪ (0, ∞)

Problem 9: Determine the domain and range of f(x) = \frac{1}{\sqrt{x}}.

Solution:

x > 0

Domain: The square root is defined for non-negative values, and the denominator cannot be zero:

Domain = (0, ∞)

Range: The function produces positive outputs that approach infinity as x approaches 0 and approach 0 as x increases.

Range = (0, ∞)

Problem 10: Find the domain and range of f(x) = sin⁡(x).

Solution:

Domain: The sine function is defined for all real numbers.

Domain = ( − ∞, ∞)

Range: The sine function oscillates between -1 and 1.

Range = [ − 1, 1]

Worksheet: Domain and Range

Worksheet for domain and range is:

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