Dominio de una funcion SECUNDARIA (4ºESO) matematicas radicales raices logaritmica

Understanding Domains in Functions

Introduction to Domains

• The instructor welcomes students and introduces the topic of domains, emphasizing its importance for upcoming exams.

• The concept of domain is crucial from 4th grade onwards, as it helps in finding vertical asymptotes.

Definition of Domain

• The domain consists of all x-values that can be substituted into a function to ensure it exists.

• For polynomial functions like x^3, the domain includes all real numbers since any number can be cubed.

Rational Functions and Denominators

• In rational functions, the denominator cannot equal zero; otherwise, it leads to indeterminate forms.

• To find restrictions on x-values, set the denominator equal to zero and solve for x.

Example with Quadratic Equations

• An example involves solving an incomplete quadratic equation where x^2 = 1.

• The resulting values (1 and -1) indicate points where the function is undefined due to division by zero.

Understanding Roots in Functions

• When dealing with square roots, the radicand must be non-negative; this applies to even-indexed roots.

• For cubic roots, all real numbers are valid since they can yield both positive and negative results.

Finding Domain through Inequalities

Setting Up Inequalities

• To determine when x^2 - 1 geq 0, we analyze intervals created by critical points (-1 and 1).

Testing Intervals for Validity

• By testing values within each interval (-∞ to -1, -1 to 1, and 1 to ∞), we identify where the expression is negative or positive.

Conclusion on Domain from Roots

• The final domain excludes values between -1 and 1 but includes -1 and 1 themselves due to their validity in square root calculations.

Domain Restrictions with Denominators

Adjusting Inequalities for Denominators

• When a root appears in the denominator, adjust inequalities from geq to >, as division by zero remains undefined.

Understanding Domain Restrictions in Functions

Vertical Asymptotes and Domain Exclusions

• The domain of the function excludes -1 and 1, as these values lead to undefined results (division by zero).

• When x is -1, the calculation yields a square root of 0, which is valid; however, division by zero occurs when evaluating the function at this point.

• The placement of values in either the numerator or denominator significantly alters the behavior of the function.

Logarithmic Functions and Their Constraints

• Logarithms cannot be computed for negative numbers; thus, expressions like x^2 - 1 must be greater than 0 to ensure validity.

• Unlike roots, logarithmic functions cannot include zero within their arguments since log(0) is considered negative infinity.

Summary and Further Assistance

• The domain restrictions discussed mirror those from previous examples. If students encounter more complex problems or anticipate challenging exam questions, they are encouraged to request additional resources or explanations.