Що таке раціональний вираз?

Rational functions are a special subset of functions that are quotients of polynomials. Thus, if $P(x)$ and $Q(x)$ are polynomials, then $\frac{P(x)}{Q(x)}$ is a rational function.

The one exception is where $Q(x) = 0$. In this case, we are dividing a polynomial by $0$, which yields in an undefined quantity.

Another interesting property of rational functions is that all polynomials are rational functions. This is because all polynomials are quotients of two polynomials:

$P(x) = \frac{P(x)}{1}$

Remember that $1$ is itself a polynomial.

Evaluating Rational Expressions

by Daytona State College Instructional Resources

In this video by Daytona State College Instructional Resources, you can walk through two examples of how to evaluate a rational expression.

Summary

Example 1: Evaluate the following rational expression: $\frac{x+7}{x-2}$ for the following values of $x$: $x=5$, $x=0$, and $x=2$.

To evaluate any type of expression, we will substitute all of the instances of $x$ with the given value.

For $x=5$,

$$\begin{align} \frac{x+7}{x-2} \\ &= \frac{5+7}{5-2} \\ &= \frac{12}{3} \\ &= 4 \\ \end{align}$$

For $x=0$,

$$\begin{align} \frac{x+7}{x-2} \\ &= \frac{0+7}{0-2} \\ &= \frac{7}{-2} \\ &= -\frac{7}{2} \\ \end{align}$$

For $x=2$,

$$\begin{align} \frac{x+7}{x-2} \\ &= \frac{2+7}{2-2} \\ &= \frac{9}{0} \\ &= \text{ undefined} \\ \end{align}$$

Evaluating Rational Functions

In this lesson, we are going to learn how to evaluate rational functions. A rational function is a fraction where both the top and the bottom are polynomials.

To evaluate a rational function, you plug in a number for $x$ and then simplify.

Let's dive deeper into the example from the image above. If we know that $f(x)=\frac{2x+1}{x-3}$, then what is $f(4)$?

First, we plug in $4$:

$$f(4)=\frac{2(4)+1}{(4)-3}$$

When we solve, we find that the answer is $9$.

Always check that the denominator isn't zero. We can't divide by zero!

Let's say in the above example that we wanted to find $f(3)$. If $x=3$, then the denominator would become zero, so $f(3)$ is undefined.

In summary: plug in the number, simplify, and make sure you don't divide by zero.

Looking Further

We'll cover the following topics in the next few lessons:

• Adding and subtracting rational expressions
• Multiplying and dividing rational expressions
• Simplifying complex rational expressions