Math Problem Statement

Determine the domain of the function f(x) = ln(81 - x^2) and identify the correct statements about f and its domain.

Solution

Let's work through the problem step by step.

Domain of the function f(x)=ln(81x2)f(x) = \ln(81 - x^2):

The natural logarithm function ln(x)\ln(x) is only defined when its argument is positive. This means we need the expression inside the logarithm, 81x281 - x^2, to be greater than zero. In other words, the domain is determined by solving the inequality:

81x2>081 - x^2 > 0

This simplifies to:

81>x281 > x^2

or equivalently:

x2>81orx2<81-x^2 > -81 \quad \text{or} \quad x^2 < 81

Taking the square root of both sides:

9<x<9-9 < x < 9

So the domain of the function f(x)f(x) is (9,9)(-9, 9).

Checking the Statements:

  1. Statement 1: ln(x) is continuous for all x>0, so f is continuous wherever 81x2>0.\text{ln}(x) \text{ is continuous for all } x > 0, \text{ so } f \text{ is continuous wherever } 81 - x^2 > 0.

    • True: Since ln(x)\ln(x) is defined and continuous for x>0x > 0, the function is continuous where 81x2>081 - x^2 > 0, which is 9<x<9-9 < x < 9.
  2. Statement 2: f is continuous for all real numbers.f \text{ is continuous for all real numbers.}

    • False: The domain of f(x)f(x) is limited to (9,9)(-9, 9). It is not defined for all real numbers.
  3. Statement 3: f is continuous on its domain.f \text{ is continuous on its domain.}

    • True: f(x)f(x) is continuous on (9,9)(-9, 9) since within this interval, 81x2>081 - x^2 > 0, and the logarithm function is continuous for positive inputs.
  4. Statement 4: f is continuous wherever 81<x2.f \text{ is continuous wherever } 81 < x^2.

    • False: This statement is misleading. The function is actually defined where 81>x281 > x^2 (not the other way around), which is 9<x<9-9 < x < 9.
  5. Statement 5: The polynomial 81x2 is continuous everywhere.\text{The polynomial } 81 - x^2 \text{ is continuous everywhere.}

    • True: The expression 81x281 - x^2 is a polynomial, and all polynomials are continuous for all real numbers. However, the logarithm limits the domain of f(x)f(x).

Conclusion:

  • The domain of the function is (9,9)(-9, 9).
  • The correct statements are: 1, 3, and 5.

Would you like any additional details or clarification?

5 Relative Questions:

  1. How do you determine the domain of a logarithmic function in general?
  2. What are the key properties of logarithmic functions regarding continuity?
  3. How would the domain change if the argument of the logarithm was x281x^2 - 81 instead of 81x281 - x^2?
  4. Can the function f(x)=ln(81x2)f(x) = \ln(81 - x^2) have any vertical asymptotes? If so, where?
  5. How does the function behave near the endpoints of the interval (9,9)(-9, 9)?

Tip:

Always remember to check the argument of the logarithm function. It must be positive for the function to be defined!

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Math Problem Analysis

Mathematical Concepts

Logarithmic Functions
Inequalities
Continuity

Formulas

ln(x) is defined for x > 0
81 - x^2 > 0

Theorems

Continuity of logarithmic functions
Continuity of polynomials

Suitable Grade Level

Grades 10-12