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

Which of the following functions have graphs that contain no asymptotes? Select all that apply.

power; y=x^n
reciprocal; y=1/x
exponetial; y=b^x
logarithmic; y=logb x
root; y=n sqr x

Identify the key characteristics of the parent fifth-root function f(x) 5 sqr 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.

What transformations produce the graph of from the graph of the parent function ? Select all that apply. (1 point)
reflection over the x-axis
reflection over the y-axis
horizontal stretch by a factor of 3
horizontal compression by a factor of 3
vertical stretch by a factor of 3
vertical compression by a factor of 3

When all four transformation types are applied to the same function, which one should be applied third? (1 point)
vertical shift
reflection
horizontal shift
stretch and compression

Describe the transformations that produce the graph of g(x)=1/2(x-4)^3+5 from the graph of the parent function f(x)=x^3. Give the order in which they must be performed to obtain the correct graph.

Answer 1

The functions that are odd are:
- f(x)=x^3+4x

The functions that have graphs that contain no asymptotes are:
- power; y=x^n
- exponential; y=b^x
- logarithmic; y=logb x

The key characteristics of the parent fifth-root function f(x) = 5√x are:
- Domain: All non-negative real numbers [0, ∞)
- Range: All non-negative real numbers [0, ∞)
- Intervals of increase: The function is increasing for all x>0
- Intervals of decrease: The function is decreasing for all x<0
- Intervals of constant: None
- x-intercept: (0,0)
- y-intercept: (0,0)
- Even or Odd: The function is neither even nor odd
- Continuity: The function is continuous for all x in the domain
- Asymptotes: There are no asymptotes
- End behavior: As x approaches infinity, the function approaches infinity. As x approaches negative infinity, the function approaches negative infinity.

The transformations that produce the graph of g(x)=1/2(x-4)^3+5 from the graph of the parent function f(x)=x^3 are:
1. Horizontal shift to the right by 4 units
2. Vertical stretch by a factor of 1/2
3. Vertical shift up by 5 units

The correct order in which the transformations must be performed is:
1. Horizontal shift
2. Vertical stretch
3. Vertical shift

Answer 2

Which of the following functions are odd? Select two answers.
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

Answer 3

To determine if a function is odd, we need to check if f(x) = -f(-x) for all x in the domain.

Let's evaluate each of the given functions:

1) f(x) = x/x^8+8x^4-7
To check if this function is odd, we substitute -x into the function:
f(-x) = (-x)/(-x)^8 + 8(-x)^4 - 7 = -x/x^8 + 8x^4 - 7
Since -x/x^8 + 8x^4 - 7 is not equal to -f(x) = -(x/x^8 + 8x^4 - 7), the first function is not odd.

2) f(x) = x^3 + 4x
To check if this function is odd, we substitute -x into the function:
f(-x) = (-x)^3 + 4(-x) = -x^3 - 4x
Since -x^3 - 4x is equal to -f(x) = -(x^3 + 4x), the second function is odd.

3) f(x) = 3x^5 + 6x^4 + 4
To check if this function is odd, we substitute -x into the function:
f(-x) = 3(-x)^5 + 6(-x)^4 + 4 = -3x^5 + 6x^4 + 4
Since -3x^5 + 6x^4 + 4 is equal to -f(x) = -(3x^5 + 6x^4 + 4), the third function is odd.

4) f(x) = 8x^2 - 8
To check if this function is odd, we substitute -x into the function:
f(-x) = 8(-x)^2 - 8 = 8x^2 - 8
Since 8x^2 - 8 is equal to -f(x) = -(8x^2 - 8), the fourth function is odd.

Therefore, the functions f(x) = x^3 + 4x and f(x) = 3x^5 + 6x^4 + 4 are odd.