Math Problem Statement

Let f(x) = (12x)/(x^2 + 16). Find the domain of f. Write your answer using interval notation.

Solution

The function given is:

f(x)=12xx2+16f(x) = \frac{12x}{x^2 + 16}

To find the domain of f(x)f(x), we need to determine where the function is defined. The main restriction for the domain of rational functions (functions that involve a fraction) comes from the denominator: it must not be zero, because division by zero is undefined.

Step 1: Identify the denominator

The denominator of f(x)f(x) is x2+16x^2 + 16.

Step 2: Check where the denominator equals zero

Set the denominator equal to zero and solve for xx:

x2+16=0x^2 + 16 = 0

x2=16x^2 = -16

Since no real number squared gives a negative result, there are no real solutions to this equation. Therefore, the denominator is never zero for any real number.

Step 3: Conclusion

Since the denominator is never zero for any real value of xx, the function is defined for all real numbers.

Domain in interval notation:

The domain of f(x)f(x) is all real numbers, which is written in interval notation as:

(,)(-\infty, \infty)

Would you like further explanation or details?


Here are five related questions to consider:

  1. How do you find the domain of a function with square roots or logarithms?
  2. What happens if the denominator of a rational function has real solutions?
  3. How would the domain change if the denominator were x216x^2 - 16?
  4. Can imaginary numbers affect the domain in real-number functions?
  5. How does the presence of absolute value functions impact domain restrictions?

Tip: Always check the denominator and square roots for restrictions when finding the domain of a function!

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Functions
Domains of Functions

Formulas

Rational function formula: f(x) = P(x)/Q(x)
Solving quadratic equations

Theorems

Division by zero is undefined

Suitable Grade Level

Grades 9-11