Rational equation

Rational function

The zeros of the rational function are determined by the zeros of the numerator. At the zeros of the denominator, the function has a point of discontinuity, and the function is not defined.

Example 1

Find the domain of the function f.

Domain of a function

The domain of a function is the values of the variable x with which it is possible to calculate the value of the function.

Solution

We can't divide by zero, so the function is not defined by the values of the variable x where the denominator is 0.

Domain restrictions

So the domain is real numbers excluding the number -2

The graph of a function consists of two parts and is not defined in -2.

The zeros of a function are determined only by the zeros of the numerator.

Example 2

Find the zeros of the function f

Solution

Equation for zeros

Domain restrictions

Zeros of the function are the zeros of the numerator

We can read the zeros also from the graph of the function.

Simplification of a rational expression

Fractions can be reduced if both the numerator and the denominator are in the form of multiplication and have common factors.

This, of course, works with all rational expressions.

If we can write the numerator and denominator in multiplication form, we can reduce the fraction. For fractional functions, this means that the numerator and denominator have at least one common zero point.

Example 3

Reduce the rational expression

Solution

We write the numerator and denominator by their factors.

Example 4

Reduce the expression for the function f and draw its graph.

Domain restrictions x ≠ 3

Zeros of the numerator x = 3 or x = –1

The graph of the function is an ascending line with a hole at x = 3. This is a single point, so when drawing a line with GeoGebra, for example, the hole is not visible. An open point is marked separately in the image.

Rational equation

When solving a rational equation, domain restrictions must be taken into account. The same tools apply to handling rational expressions as with fractions.

Example 5

Solve

The domain restrictions for the equation are obtained from the zeros of the denominators .: x ≠ –2 and x ≠ 1

We simplify the equation by expanding rational expressions

Expand to the same name.
Add the fractional expressions.
Open the brackets and simplify.
Multiply the equation by the denominator. The denominator is different from zero.
A quadratic equation is obtained with solutions

Both solutions are valid as they are not included in the domain restrictions.

Example 6

Find the zeros of the function f

Domain restrictions x ≠ –1 and x ≠ 1

We set the function equal to zero and solve the equation

Solutions to quadratic equation

Of these, only x = 2 are not in the domain restrictions. The zero point is x = 2.

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