Math Problem Statement

How many of the functions listed below are correctly defined? f : (0, ∞) → ℝ, f(x) = cos((x^2 - 1)/x), g : ℝ \ {2} → ℝ, g(x) = ln((x^2 - 4)/(x - 2)), h : [0, ∞) → ℝ, h(x) = √(x^2 + 4)

Solution

Let's analyze the functions given in the image to determine how many are correctly defined.

Function 1: f:(0,)R,f(x)=cos(x21x)f : (0, \infty) \to \mathbb{R}, \quad f(x) = \cos\left(\frac{x^2 - 1}{x}\right)

  • The domain is (0,)(0, \infty), meaning xx must be positive.
  • The expression inside the cosine function is x21x\frac{x^2 - 1}{x}, which simplifies to x1xx - \frac{1}{x}. This expression is defined for x>0x > 0 (no issues with division by zero).
  • The cosine function is always defined for any real input.

Thus, f(x)f(x) is correctly defined on (0,)(0, \infty).

Function 2: g:R{2}R,g(x)=ln(x24x2)g : \mathbb{R} \setminus \{2\} \to \mathbb{R}, \quad g(x) = \ln\left(\frac{x^2 - 4}{x - 2}\right)

  • The domain excludes x=2x = 2 because of the denominator x2x - 2.
  • The expression simplifies as: x24x2=(x2)(x+2)x2=x+2forx2\frac{x^2 - 4}{x - 2} = \frac{(x - 2)(x + 2)}{x - 2} = x + 2 \quad \text{for} \quad x \neq 2 So, g(x)=ln(x+2)g(x) = \ln(x + 2).
  • For g(x)g(x) to be defined, the argument of the logarithm x+2>0x + 2 > 0, or x>2x > -2.

Thus, the domain should be (2,2)(2,)(-2, 2) \cup (2, \infty), but the function is undefined for x2x \leq -2. Since the domain given is R{2}\mathbb{R} \setminus \{2\}, which includes values less than 2-2, g(x)g(x) is not correctly defined.

Function 3: h:[0,)R,h(x)=x2+4h : [0, \infty) \to \mathbb{R}, \quad h(x) = \sqrt{x^2 + 4}

  • The expression inside the square root is x2+4x^2 + 4, which is always positive for any real xx, so the square root is defined for all x0x \geq 0.
  • The domain is [0,)[0, \infty), which fits the condition for the square root function.

Thus, h(x)h(x) is correctly defined.

Conclusion:

  • f(x)f(x) and h(x)h(x) are correctly defined, while g(x)g(x) is not.
  • Therefore, 2 functions are correctly defined.

The correct answer is 2.

Would you like a detailed explanation of any part of this?

Here are some related questions:

  1. How do we simplify rational expressions?
  2. Why do logarithmic functions have domain restrictions?
  3. What are the general rules for the domain of square root functions?
  4. How does the cosine function behave for different inputs?
  5. What are the properties of logarithmic functions?

Tip: Always check the domain of the input inside special functions like logarithms and square roots, as they often introduce restrictions.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Function domains
Cosine function
Logarithmic functions
Square root functions
Algebraic simplifications

Formulas

cos(x) for trigonometric functions
ln(x) for logarithmic functions
sqrt(x) for square roots

Theorems

Rules for domains of logarithmic functions
Square root and real numbers theorem

Suitable Grade Level

Undergraduate level (introductory calculus or precalculus)