How to Find Domain and Range of a Function?

The domain and range of a function is the set of all possible inputs and outputs of a function respectively. Here you get familiarized with the domain and range of a function and how to calculate it.

How to Find Domain and Range of a Function?

The domain and range of any function can be found algebraically or graphically.

Step by Step guide to finding domain and range

The domain and range are defined for a relation and they are the sets of all the \(x\)-coordinates and all the \(y\)-coordinates of ordered pairs respectively. For example, if the relation is,\(R = {(1, 2), (2, 2), (3, 3), (4, 3)}\), then:

  • Domain \(=\) the set of all \(x\)-coordinates \(= {1, 2, 3, 4}\)
  • Range \(=\) the set of all \(y\)-coordinates \(= {2, 3}\)

Domain and range of a function

The domain and range of a function are the components of a function. The domain is the set of all the input values of a function and the range is the possible output given by the function.

\(\color{blue}{Domain→ Function →Range}\)

If there exists a function \(f: A →B\) such that every element of \(A\) is mapped to elements in \(B\), then \(A\) is the domain, and \(B\) is the co-domain. The image of an element \(a\) under a relation \(R\) is given by \(b\), where \((a,b) ∈ R\). The range of the function is the set of images. The domain and range of a function is denoted in general as follows: Domain \((f) = {x ∈ R}\) and range \((f)={f(x) : x ∈ domain\: (f)}\).

The domain and range of this function \(f(x) = 2x\) is given as domain \(D ={x ∈ N }\) , range \(R = {(y): y = 2x}\).

Domain of a function

The domain of a function refers to “all the values” that go into a function. Here are the general formulas used to find the domain of different types of functions. Here, R is the set of all real numbers:

  • The domain of any polynomial (linear, quadratic, cubic, etc) function is \(\color{blue}{R}\).
  • The domain of a square root function \(\sqrt{x}\) is \(\color{blue}{x≥0}\).
  • The domain of an exponential function is \(\color{blue}{R}\).
  • The domain of the logarithmic function is \(\color{blue}{x>0}\).
  • To find the domain of a rational function \(y = f(x)\), set the \(\color{blue}{denominator ≠ 0}\).

Range of a function

The range of a function is the set of all its outputs. Example: Let’s consider a function \(f: A→ B\), where \(f(x) = 2x\) and each of \(A\) and \(B =\) {set of natural numbers}. Here we say \(A\) is the domain and \(B\) is the co-domain. Then the output of this function becomes the range. The range \(=\) {set of even natural numbers}. Domain elements are called pre-images and the elements of the co-domain which are mapped are called the images. Here, the range of the function \(f\) is the set of all images of the elements of the domain (or) the set of all the outputs of the function.

Here are the general formulas used to find the range of different types of functions. Note that \(R\) is the set of all real numbers here.

  • The range of a linear function is \(\color{blue}{R}\).
  • The range of a quadratic function \(y=a(x-h)^2+k\) is: if \(\color{blue}{a>0, y≥k}\), and  if \(\color{blue}{a<0, y≤k}\).
  • The range of a square root function is \(\color{blue}{y≥0}\).
  • The range of an exponential function is \(\color{blue}{y>0}\).
  • The range of the logarithmic function is \(\color{blue}{R}\).
  • To find the range of a rational function \(y = f(x)\), solve it for \(x\) and set the \(\color{blue}{denominator ≠ 0}\).

Domain and range of trigonometric functions

Look at the graph of the \(sin\) and the \(cos\) function. Note that the value of the functions oscillates between \(-1\) and \(1\) and is defined for all real numbers.

Thus, for each of the \(sin\) and \(cos\) functions:

  • Domain: The domain of the functions is the set \(\color{blue}{R}\).
  • Range: The range of the functions is \(\color{blue}{[-1, 1]}\).

The domain and range of all trigonometric functions are shown below:

Domain and range of an absolute value function

The function \(y=|ax+b|\) is defined for all real numbers. So, the domain of the absolute value function is the set of all real numbers. The absolute value of a number always results in a non-negative value. Thus, the range of an absolute value function of the form \(y= |ax+b|\) is \(y ∈ R | y ≥ 0\).

The domain and range of an absolute value function are given as follows:

  • Domain \(\color{blue}{= R}\)
  • Range \(\color{blue}{= [0, ∞)}\)

Domain and range of a square root function

The function \(y=\sqrt{\left(ax+b\right)}\) is defined only for \(x ≥ -\frac{b}{a}\). So, the domain of the square root function is the set of all real numbers greater than or equal to \(\frac{b}{a}\). We know that the square root of something always results in a non-negative value. Thus, the range of a square root function is the set of all non-negative real numbers.

The domain and range of a square root function are given as follows:

  • Domain \(\color{blue}{= [-\frac{b}{a},∞)}\)
  • Range \(\color{blue}{= [0,∞)}\)

Graphs of domain and range

Another way to identify the domain and range of functions is to use graphs. Domain refers to the set of possible input values. The domain of a graph contains all the input values shown on the \(x\)-axis. The range is the set of possible output values shown on the \(y\)-axis. The easiest way to find the range of a function is to plot it and search for the \(y\)-values covered by the graph.

To find the quadratic function range, it suffices to see whether it has a maximum or minimum value. The maximum/minimum value of a quadratic function is the \(y\)-coordinate of its vertex. To find the domain of the rational function, set the denominator as \(0\) and solve for the variable.

The domain is denoted by all the values from left to right along the \(x\)-axis and the range is given by the span of the graph from the top to the bottom.

Finding Domain and Range – Example 1:

Find the domain and range of the function \(y=2-\sqrt{\left(-3x+2\right)}\).

Solution:

A square root function is defined only when the value inside it is a non-negative number. So for a domain,

\(-3x+2≥0\)

\(-3x≥-2\)

\(x≤\frac{2}{3}\)

We know that the square root function results in a non-negative value always. So for a range,

\(\sqrt{\left(-3x+2\right)}\ge 0\)

\(-\sqrt{\left(-3x+2\right)}\le \:0\)

\(2-\sqrt{\left(-3x+2\right)}\le \:2\)

\(y\le \:2\)

Exercises for Finding Domain and Range

Find the domain and range of the function.

  1. \(\color{blue}{f(x)=\frac{x+1}{3-x}}\)
  2. \(\color{blue}{f\left(x\right)=e^x+4}\)
  3. \(\color{blue}{f\left(x\right)=|7-2x|}\)
  4. \(\color{blue}{f\left(x\right)=4-x^2}\)
  5. \(\color{blue}{f\left(x\right)=-\frac{7}{x}}\)
This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{D=\left(-\infty \:,3\right)\cup \left(3,\infty \:\right), R=\left(-\infty ,-1\right)\cup \:\left(-1,\infty \right)}\)
  2. \(\color{blue}{D=\left(-\infty ,\infty \right), R=\left(4,\infty \right)}\)
  3. \(\color{blue}{D=\left(-\infty ,\infty \right), R=[0,\infty)}\)
  4. \(\color{blue}{D=\left(-\infty ,\infty \right), R=(-\infty,4]}\)
  5. \(\color{blue}{D=\left(-\infty \:,0\right)\cup \left(0,\infty \:\right), R=\left(-\infty \:,0\right)\cup \left(0,\infty \:\right)}\)

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