To find the domain of the rational function $$R(x) = \frac{2x^2 - 10x}{4x^2 - 16x - 20}$$, we follow the procedure to determine the domain of a rational function by finding and excluding values that cause division by zero.

Step 1. Set the denominator equal to zero: $$4x^2 - 16x - 20 = 0$$.

Step 2. Solve the equation. First, factor out the greatest common factor: $$4(x^2 - 4x - 5) = 0$$. Next, factor the trinomial: $$4(x - 5)(x + 1) = 0$$. Using the Zero Product Property, set each variable factor to zero: $$x - 5 = 0$$ or $$x + 1 = 0$$. Solving these equations yields $$x = 5$$ and $$x = -1$$.

Step 3. The domain is all real numbers excluding those values found in Step 2. Therefore, the domain of $$R(x)$$ is all real numbers where $$x 
eq 5$$ and $$x 
eq -1$$.

Example of Determining the Domain of the Rational Function $$R(x) = \frac{2x^2 - 10x}{4x^2 - 16x - 20}$$

To find the domain of the rational function $$R(x) = \frac{4x^2 - 16x}{8x^2 - 16x - 64}$$, determine which values make the denominator equal to zero and exclude them.

Step 1. Set the denominator equal to zero: $$8x^2 - 16x - 64 = 0$$.

Step 2. Solve the equation. Begin by factoring out the greatest common factor: $$8(x^2 - 2x - 8) = 0$$. Then, factor the trinomial: $$8(x - 4)(x + 2) = 0$$. Apply the Zero Product Property by setting each variable factor to zero: $$x - 4 = 0$$ or $$x + 2 = 0$$. Solving these equations gives $$x = 4$$ and $$x = -2$$.

Step 3. State the domain by excluding the values found in Step 2. Thus, the domain of $$R(x)$$ is all real numbers where $$x 
eq 4$$ and $$x 
eq -2$$.

Example of Determining the Domain of the Rational Function $$R(x) = \frac{4x^2 - 16x}{8x^2 - 16x - 64}$$

To determine the domain of a rational function, follow this three-step procedure:

Step 1. Set the denominator expression equal to zero.
Step 2. Solve the resulting equation to find the values that make the denominator zero.
Step 3. State the domain as the set of all real numbers excluding the specific values found in Step 2.

Claude

To determine the specific values for which a rational expression is undefined, follow these steps:
1. Set the expression in the denominator equal to $$0$$.
2. Solve the resulting equation for the variable.

The solutions obtained from this equation represent the values that make the original rational expression undefined, as they would result in division by zero.

Procedure for Determining Undefined Rational Expressions

https://openstax.org/details/books/intermediate-algebra-2e

OpenStax Intermediate Algebra

To determine the values for which the rational expression $$\frac{x + 4}{x^2 + 5x + 6}$$ is undefined, follow the procedure by setting the denominator equal to $$0$$ and solving for the variable $$x$$:

$$x^2 + 5x + 6 = 0$$

Factor the quadratic equation:
$$(x + 2)(x + 3) = 0$$

Set each factor to zero:
$$x + 2 = 0$$ or $$x + 3 = 0$$
$$x = -2$$ or $$x = -3$$

Thus, the rational expression is undefined for the values $$x = -2$$ and $$x = -3$$.

Learn Before

Related

Learn After