Irrational functions are generally considered as functions that contain the radical sign. For example, functions that contain square roots, cube roots, or other roots are considered irrational functions.

Here, we will look at a summary of irrational functions along with their most important characteristics. Also, we will explore various examples with answers to learn how to use these functions.

## Summary of irrational functions

There is no rigorous definition of irrational functions. We can say that an irrational function is a function that cannot be written as the quotient of two polynomials, but this definition is not commonly used.

Generally, the most used definition is that an irrational function is a function that includes variables in radicals, that is, square roots, cubic roots, and others. Therefore, the fundamental form of an irrational function is:

$latex \sqrt[n]{g(x)}$

where, $latex g(x)$ is a rational function.

- If the index
*n*of the radical is odd, it is possible to calculate the domain of all real numbers. - If the index
*n*of the radical is even, we need $latex g(x)$ to be positive or zero since the even roots of a negative number are not real.

The following is a graphical representation of the irrational function $latex \sqrt{5-x}$:

## Examples with answers of irrational functions

In the following examples, we can look at how to work with irrational functions. These functions are usually a bit more difficult to manipulate, but it is recommended that you try to solve the exercises yourself before looking at the answer.

**EXAMPLE 1**

If we have $latex f(x)=\sqrt{x-6}$, find the value of $latex f(10)$.

##### Solution

To evaluate a function, we simply have to use the given input value:

$latex f(10)=\sqrt{10-6}$

$latex =\sqrt{4}$

$latex =2$

**EXAMPLE 2**

If we have $latex f(x)=-\sqrt{5-x}$, find the value of $latex f(2)$.

##### Solution

Again, we simply have to use the given input value:

$latex f(2)=-\sqrt{5-2}$

$latex =-\sqrt{3}$

$latex \approx -1.732$

**EXAMPLE 3**

Graph the function $latex f\left( x \right)=\sqrt{{2-x}}$

##### Solution

We have to start by identifying the domain to know for which values of *x* we are restricted. We know that it is not possible to have negative numbers within square roots. For example, if we had $latex x=4$, we would get the following:

$latex f\left( x \right)=\sqrt{{2-4}}=\sqrt{{-2}}$

We can find the domain by taking the expression that is inside the radical and forming an inequality with “equal to or greater than zero” and solving for *x*:

$latex 2-x\ge 0$

$latex 2\ge x$

$latex x\le 2$

Therefore, we should not choose any value of *x* that is greater than 2 for the table of values that we will form. Forming a table with several values of *x*, we have:

Now, we can graph these points on the Cartesian plane:

We draw a curve that passes through these points without extending the graph to the right of $latex x=2$:

We should not draw straight lines as these functions produce curves. It is advisable to take values of *x* that are separate from each other to get better reference points.

**EXAMPLE 4**

Graph the function $latex f\left( x \right)=-\sqrt{{2x-3}}$.

##### Solution

Again, we have to start by finding the domain of the function, then we form an inequality to get non-negative values in the expression inside the radical:

$latex 2x-3\ge 0$

$latex 2x\ge 3$

$latex x\ge \frac{3}{2}$

The first point on the graph will be at $latex x=\frac{3}{2}$, so we will start from there and take larger values of *x*:

After graphing the points and drawing the curve that passes through those points, we obtain the following graph:

**EXAMPLE 5**

Graph the function $latex f\left( x \right)=2\sqrt{{3x+2}}-1$.

##### Solution

We form an inequality with the expression inside the radical to find the domain of the function:

$latex 3x+2\ge 0$

$latex 3x\ge -2$

$latex x\ge \frac{-2}{3}$

We start with the point $latex x=\frac{-2}{3}$, we take several values of *x* to obtain points that are located on the curve:

Plotting the points on the Cartesian plane and drawing the curve, we obtain the following graph:

The graph grows to the right indefinitely, however, the domain is restricted to $latex x \ge \frac{-2}{3}$ so we have no more values to the left and the graph starts at the point $latex (\frac{-2 }{3}, ~-1)$.

## See also

Interested in learning more about functions? Take a look at these pages:

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