Domain and Range for Graph - GeeksforGeeks (2024)

Domain and Range are fundamental in understanding the behavior of functions when they are represented graphically. Domain of a function refers to all the possible input values (typically represented on the x-axis) that the function can accept without resulting in any undefined behavior. On the other hand, the range is the set of all possible output values (typically shown on the y-axis) that the function can produce.

In this article, we will discuss all things related to Domain and Range for Graph

Domain and Range

Domain and range of any particular function are fundamental concepts to understand about functions in mathematics, and especially when dealing with graphs of functions. The domain of a function can be defined as set of all possible x-values that can be taken by the function while the range can be defined as set of all possible y-values that can be taken by the function.

While this idea may not have direct applications in daily life, it is important for students because based on this concept, functions may be analyzed or interpreted in numerous fields like physics, engineering, economics and others.

What is the Domain?

The domain of a function is the complete set of possible values of the independent variable, which are typically the x-values.

How to Find the Domain from a Graph?

Finding the domain of a function from its graph involves identifying all the possible input values (x-values) for which the function is defined. Here’s a step-by-step approach:

Step 1: Look at the range of x-values over which the function exists.

Step 2: Check if there are any breaks, holes, or vertical asymptotes in the graph. These indicate x-values where the function is not defined.

Step 3: For graphs with endpoints (e.g., closed intervals), include these x-values if they are part of the domain.

Step 4: For functions that extend infinitely, determine if the domain extends to positive or negative infinity.

Step 5: Write the domain in interval notation, set-builder notation, or as a union of intervals, depending on the graph’s characteristics.

Example for Domain

• Linear Function: For f(x) = 2x + 3 the graph is a straight line extending infinitely in both direction hence the set of all real numbers is included in the domain which is (-∞, ∞).
• Square Root Function: If f(x) = x, The begin at x=0 on the number line and goes on to the right. It follows that for the domain of the variable x, it means it is greater than equal to 0, that is [0, ∞).

What is the Range?

Range refers to the set of all possible output values (or y-values) that a function can produce. In other words, it is the set of all values that the function maps to from the domain (the set of input values or x-values).

To find the range of a function, you consider all possible values that the function can take as you vary the input values over the domain.

For a simple linear function like f(x) = 2x + 3, where x can be any real number:

• The range is also all real numbers because, as x varies, f(x) can take any real value.

How to Find the Range from a Graph?

Finding the range of a function from its graph involves identifying the set of all possible output values (y-values) that the function takes. Here are the steps for the same:

Step 1: Look at the lowest point and the highest point that the graph reaches along the y-axis. These points represent the minimum and maximum values that the function attains (if they exist).

Step 2: Check if the graph extends indefinitely in the vertical direction. For example, if the graph continues to rise or fall without bound, then the range will be all real numbers (for instance, −∞ to ∞).

Step 3:

• For a graph that opens upward or downward (like a parabola), the range might start from the vertex and extend infinitely in one direction.
• For periodic functions like sine or cosine, the graph repeats in a regular pattern, and the range is determined by the highest and lowest points in one cycle.

Step 4: If there are any gaps in the graph (e.g., a function with asymptotes or holes), then those y-values are excluded from the range.

Step 5: Based on your observations, express the range. Use interval notation if possible:

• If the function has a minimum value a and no maximum, the range might be [a,∞).
• If it has both a minimum and maximum, the range could be [a, b].
• If the graph covers all possible y-values, the range is (−∞,∞).

Examples of Range

• Quadratic Function: If the function is f(x) = x², the graph formed is a parabola, which is a up- ward opening parabola with minimum value of y = 0. As such, the interval of values is the set of all non-negative, infinitive numbers, or [0, ∞).
• Trigonometric Function (Sine): This is because the absolute largest value a sine function may reach is 1 and the absolute minimum value is -1 due to the nature of the sine function; from period to period, the graph go up to 1 and down to -1 giving the range [−1, 1].

Graphical Representation of Domain and Range

To graphically represent the domain and range of a function, you typically follow these steps:

• First, you need to plot the function on a coordinate plane.
• Domain of a function consists of all the possible input values (x-values) for which the function is defined.
• Range of a function consists of all the possible output values (y-values) that the function can produce.

Visualizing Domain on the X-Axis

To visualize a domain on the x-axis, typically in a graph or plot, the x-axis represents the domain values, while the y-axis represents the range (the corresponding output values).

Visualizing Range on the Y-Axis

To visualize a range on the y-axis, typically in a graph or plot, the y-axis represents the range values, while the x-axis represents the domain (the corresponding input values).

Read More,

• Domain and Range of Function
• Range of a Function
• Domain and Range of a Relation
• Difference between Codomain and Range

Solved Problems: Domain and Range

Problem 1: Find the domain and range of the function f(x) = 1/x-2.

Solution:

Domain: The function is undefined when x equals 2 so the domain of this sil is negative infinity and 2 union two and infinity.

Range: Since the function can take any value other than zero, the values of the function at x = 0 are excluded, thus the range is minus ‘∞’ and plus ‘∞’.

Problem 2: Determine the domain and range of f(x) = (4−x²)^1/2

Solution:

Domain: The expression under the square root must be non-negative so that 4- x^2 ≥ 0. Solving −2 ≤ x ≤ 2 the domain is [−2, 2].

Range: This in turn equals to 2 only when x = 0 then greater than 0 to equal 0 when x = ±2 as so make the interval of the range from 0 up to 2.

Problem 3: Determine the domain and range of the function f(x) = x² + 3x + 2.

Solution:

Domain: The function f(x) = x² + 3x + 2 is quadratic and it has all the characteristics similar to the quadratic polynomial f(x)=ax²+bx+c whereby ?=1, ?=3 & ?=2 The function is defined for all real numbers. Thus, its domain is all the real numbers, which are indicated by the interval from negative infinity to positive infinity.

Range: To find the range we begin with the vertex of the parabola because it is the lowest or highest point about which the function opens. The vertex form of a quadratic function of the form ?x² + ?x + ? is x = −?/(2?).

For f(x) = x² + 3x + 2, we calculate:

x = k/n = -3/(2 × 1) = -3/2.

Next, we substitute x = −3/2 into the function to find the minimum value:

f(−3/2) = (−3/2)² + 3(−3/2) + 2 = 9/4 − 9/2 + 2 = 1/4.

From the fact that the parabola opens upward, we know that range is [1/4, ∞).

Problem 4: Identify the domain and range of the function f(x) = 9 − x².

Solution:

Domain: The definition of the function f(x) = 9 − x² is when the following is true: √(9−x²) becomes real, or in other words 9−x² ≥ 0.

This leads to x² ≤ 9 or in other words -3 ≤ x ≤ 3.

Hence, the domain of is [−3, 3].

Range: The function f(x) = 9 − x² is the graph of the upper half of a circle with radius 3 centred at the origin. It has a minimum value of 0 at x=3 or x=-3 and a maximum value of 3 at x=0. Thus range is [0, 3].

Problem 5: Identify the domain and range of the function f(x) = ?^x + 2.

Solution:

Domain: The exponential function ?^x is defined for all real numbers so the domain of the function f(x) = ?^x + 2 is also (−∞, ∞.

Range: The smallest value which ?^x can attain is 0 and this is attained as x moves closer to negative infinity. As such the minimum value of ?(x)= 2 and as the value of x increases the value of ?(x) increases without any upper limit. Hence the range is (2, infinity).

Problem 6: Determine the domain and range of the function f(x) = 2x^2 + 1.

Solution:

Domain: The function f(x) = 2x^2 + 1 can be defined for all the real numbers which makes the domain as (∞, ∞).

Range: The fundamental of x^2 + 1 is 1 and this is achieved when x = 0. When x^2 increases, the function increases without bound and, therefore the range is [1, ∞).

Problem 7: Find the domain and range of the function f(x) = sin(x) + 1.

Solution:

Domain: The given function f(x) = sin(x) + 1 is valid for all the real values of x so the domain of this function is (-∞ ∞).

Range: The sine function ranges between -1 and 1 and thus f(x) ranges between 0 and 2. Hence, the range is as follows: [0,2].

Problem 8: Identify the domain and range of the function f(x) = |x − 2|.

Solution:

Domain: The function f(x) = |x − 2| is defined for all real numbers, so the domain is (−∞, ∞).

Range: The absolute value function ∣x−2∣ is always non-negative and reaches its minimum value of 0 when x = 2. Therefore, the range is [0,∞).

Problem 9: Identify the domain and range of f(x)=tan(x)

Solution:

Domain: The function f(x)=tan(x) is undefined when cos(x)=0, which occurs at x = π/2 + nπ, where n is an integer. Therefore, the domain is: x∈R∖{π/2 + nπ∣n∈Z}

Range: The tangent function can take any real value, so the range is (−∞,∞).

Problem 10: Find the domain and range of f(x) = 1/x-1 for x ≥ 0

Solution:

Domain: The function f(x) = 1/x-1 is undefined when x = 1. Given that x ≥ 0, the domain is: [0, 1) ∪ (1, ∞)

Range: As x approaches 1 from the left, f(x) approaches −∞, and as x approaches 1 from the right, f(x) approaches ∞. Therefore, the range is (−∞, 0) ∪ (0, ∞).

Domain and Range: Practice Problems

Instructions: Identify the domain and range for the following functions.

Problem 1: f(x) = x² – 4x + 4

Problem 2: f(x) = 3/(x + 2)

Problem 3: f(x) = (25 – x²)^1/2

Problem 4: f(x) = log(x + 5)

Problem 5: f(x) = e^2x

Problem 6: f(x) = sin(2x)

Problem 7: f(x) = |x – 3| + 2

Problem 8: f(x) = cos(x) – 1

Problem 9: f(x) = 1/(x² – 1)

Problem 10: f(x) = (x – 1)^1/2

Answer Key:

1: Domain: (-∞, ∞), Range: [0, ∞)

2: Domain: (-∞, -2) ∪ (-2, ∞), Range: (-∞, 0) ∪ (0, ∞)

3: Domain: [-5, 5], Range: [0, 5]

4: Domain: (-5, ∞), Range: (-∞, ∞)

5: Domain: (-∞, ∞), Range: (0, ∞)

6: Domain: (-∞, ∞), Range: [-1, 1]

7: Domain: (-∞, ∞), Range: [2, ∞)

8: Domain: (-∞, ∞), Range: [-2, 0]

9: Domain: (-∞, -1) ∪ (-1, 1) ∪ (1, ∞), Range: (-∞, 0) ∪ (0, ∞)

10: Domain: [1, ∞), Range: [0, ∞)

FAQs

What is the domain and range of a function?

Domain is the collection of all elements of interest (the independent variable x) and the range is the collection of all the possible corresponding elements of interest (the dependent variable y).

How can I find the domain of a function?

To find the range, identify the y-values that the function attains and stay way from values that result in undecidedness such as division by zero and values less than 0 under a square root.

How do I determine the range of a function from a graph?

To notice the range look specifically at the lowest and the highest point marked on the graph’s y-axis.

Can a function have an infinite range?

Yes, for instance, straight lines such as identity function f(x) = x have a range of: (–∞, ∞).

What is the domain of a logarithmic function?

The domain of a logarithmic function f(x) = log(x) is (0,∞) because logarithm is not defined in case of non-positive values of x.

Why is understanding domain and range important?

About the domain and range, it is very important in the actual interpretation and analysis of functions which are fundamental in many mathematical applications.

Are there functions with the same domain but different ranges?

Yes, a function can share the same domain but the range different based on how the output values are set.