How to Find Domain and Range of Radical Functions? (+FREE Worksheet!)

There are several different methods to identify the Domain and Range of Radical Functions. In this blog post, you will learn how to find the Domain and Range of Radical Functions.

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Rules for Finding Domain and Range of Radical Functions

  • To find the domain of the function, find all possible values of the variable inside radical.
  • Remember that having a negative number under the square root symbol is not possible. (For cubic roots, we can have negative numbers)
  • To find the range, plugin the minimum and maximum values of the variable inside the radical.

Examples

Domain and Range of Radical Functions – Example 1:

Find the domain and range of the radical function. \(y=\sqrt{x-3}\)

Solution:

For domain: Find non-negative values for radicals: \(x-3≥0\)
Domain of functions: \(\sqrt{f(x)} →f(x)≥0\), Then solve \(x-3≥0→x-3+3 ≥0+3 → x≥3\)
Domain of the function \( y=\sqrt{x-3}: x≥3\)
For range: The range of a radical function of the form \(c\sqrt{ax+b}+k\) is:
\(f(x)≥k\)
For the function \(y=\sqrt{x-3}\), the value of \(k\) is \(0\). Then Range of the function \(y=\sqrt{x-3}: f(x)≥0\)

Domain and Range of Radical Functions – Example 2:

Find the domain and range of the radical function. \(y=\sqrt{x-8}+5\)

Solution:

For domain: Find non-negative values for radicals: \(x-8≥0\)
Domain of functions: \(\sqrt{f(x)} →f(x)≥0\), Then solve for \(x: x-8≥0→x-8+8 ≥ 0+8→ x≥8\)
Domain of the function \(y=\sqrt{x-8}+5:\) \(x≥8\)
For range: the range of a radical function of the form \(c\sqrt{ax+b}+k\) is \(f(x)≥k\)
For the function \(y=\sqrt{x-8}+5\), the value of \(k\) is \(5\). Then Range of the function \(y=\sqrt{x-8}+5 : f (x) ≥5\)

Domain and Range of Radical Functions – Example 3:

Find the domain and range of the radical function. \(y=\sqrt{2x+4}+9\)

Solution:

For domain: Find non-negative values for radicals: \(2x+4≥0\)
Domain of functions: \(\sqrt{f(x)} →f(x)≥0\), Then solve for \(x: 2x+4≥0→x≥-2\)
Domain of the function \(y=\sqrt{2x+4}+9:\) \(x≥-2\)
For range: the range of a radical function of the form \(c\sqrt{ax+b}+k\) is \(f(x)≥k\)
For the function \(y=\sqrt{2x+4}+9\), the value of \(k\) is \(9\). Then Range of the function \(y=\sqrt{2x+4}+9 :\) \(f(x)≥9\)

Domain and Range of Radical Functions – Example 4:

Find the domain and range of the radical function. \(y=3\sqrt{2x-10}+6\)

Solution:

For domain: Find non-negative values for radicals: \(2x-10≥0\)
Domain of functions: \(\sqrt{f(x)} →f(x)≥0\), Then solve for \(x: 2x-10≥0→x≥5\)
Domain of the function \(y=3\sqrt{2x-10}+6:\) \(x≥5\)
For range: the range of a radical function of the form \(c\sqrt{ax+b}+k\) is \(f(x)≥k\)
For the function \(y=3\sqrt{2x-10}+6\), the value of \(k\) is \(6\). Then Range of the function \(y=3\sqrt{2x-10}+6:\) \(f(x)≥6\)

Exercises for Domain and Range Radical Functions

Identify the Domain and Range

  1. \(\color{blue}{y=\sqrt{x-1}}\)
  2. \(\color{blue}{y=2\sqrt{x+3}}\)
  3. \(\color{blue}{y=\sqrt{x-5}}\)
  4. \(\color{blue}{y=\sqrt{x+4}+2}\)

Sketch the graph of the function

5. \(\color{blue}{y=\sqrt{x}-2}\)

6. \(\color{blue}{y=3\sqrt{x}-1}\)

  1. \(\color{blue}{x≥1,f(x)≥0}\)
  2. \(\color{blue}{x≥-3,f(x)≥0}\)
  3. \(\color{blue}{x≥5,f(x)≥0}\)
  4. \(\color{blue}{x≥-4,f(x)≥2}\)

5. \(\color{blue}{y=\sqrt{x}-2}\)

6. \(\color{blue}{y=3\sqrt{x}-1}\)

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