Which of the following functions are odd? Select all that apply.

Question

f(x)=x/x^8+8x^4-7
f(x)=x^3+4x
f(x)=3x^5+6x^4+4
f(x)=8x^2-8

Identify the key characteristics of the parent fifth-root function f(x)= \root(5)(x). Include the following: domain, range, intervals where the function is increasing, decreasing, or constant, intercepts, whether the function is even, odd, or neither, whether the function is continuous or discontinuous, asymptotes, and end behavior.

Answer 1

A function f(x) is odd if f(-x) = -f(x) for all x in the domain of the function.

For the given functions:
1. f(x) = x/x^8+8x^4-7: This function is neither odd nor even.
2. f(x) = x^3 + 4x: This function is odd because f(-x) = (-x)^3 + 4(-x) = -x^3 - 4x = -f(x).
3. f(x) = 3x^5 + 6x^4 + 4: This function is neither odd nor even.
4. f(x) = 8x^2 - 8: This function is even because f(-x) = 8(-x)^2 - 8 = 8x^2 - 8 = f(x).

Key characteristics of the parent fifth-root function f(x) = \root(5)(x):
- Domain: The domain is all real numbers.
- Range: The range is all real numbers.
- Intervals of increase, decrease, or constancy: The function is increasing for x > 0, decreasing for x < 0, and constant at x = 0.
- Intercepts: The x-intercept is (0, 0) and the y-intercept is (1, 1).
- Even or odd: The function is neither even nor odd.
- Continuity: The function is continuous for all real numbers.
- Asymptotes: There are no vertical asymptotes.
- End behavior: As x approaches positive or negative infinity, f(x) approaches positive or negative infinity, respectively.