Examples of Irrational Function Problems

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.

ALGEBRA
graph of irrational functions

Relevant for

Learning about irrational functions with examples.

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ALGEBRA
graph of irrational functions

Relevant for

Learning about irrational functions with examples.

See examples

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:

\sqrt[n]{g(x)}

where, 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 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 \sqrt{5-x}:

example of irrational function

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 f(x)=\sqrt{x-6}, find the value of f(10).

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

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

=\sqrt{4}

=2

EXAMPLE 2

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

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

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

=-\sqrt{3}

\approx -1.732

EXAMPLE 3

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

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 x=4, we would get the following:

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:

2-x\ge 0

2\ge x

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:

table of values of irrational function 2

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

points to plot irrational graph 2

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

graph of irrational function using points 1

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 f\left( x \right)=-\sqrt{{2x-3}}.

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:

2x-3\ge 0

2x\ge 3

x\ge \frac{3}{2}

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

table of values of irrational function 3

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

graph of irrational function using points 2

EXAMPLE 5

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

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

3x+2\ge 0

3x\ge -2

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

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

table of values of irrational function 4

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

graph of irrational function using points 3

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


See also

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

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