Which of the following functions are odd? Select all that apply.
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.
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.
To determine which functions are odd, we need to understand the concept of odd functions. An odd function is defined as a function in which the values for negative inputs are the opposite of the values for positive inputs. In other words, if f(x) is an odd function, then f(-x) = -f(x) for all values of x.
Let's check each of the given functions to see if they are odd:
1. f(x) = x/x^8+8x^4-7
To determine if this function is odd, we need to substitute -x in place of x and simplify:
f(-x) = (-x)/(-x)^8+8(-x)^4-7
Simplifying further:
f(-x) = -x/x^8+8x^4-7
Since -x/x^8+8x^4-7 is not equal to -f(x), the function is not odd.
2. f(x) = x^3+4x
Similarly, we substitute -x in place of x and simplify:
f(-x) = (-x)^3+4(-x)
Simplifying further:
f(-x) = -x^3-4x
Since -x^3-4x is equal to -f(x), the function is odd.
3. f(x) = 3x^5+6x^4+4
Substituting -x in place of x and simplifying:
f(-x) = 3(-x)^5+6(-x)^4+4
Simplifying further:
f(-x) = -3x^5+6x^4+4
Since -3x^5+6x^4+4 is not equal to -f(x), the function is not odd.
4. f(x) = 8x^2-8
Substituting -x in place of x and simplifying:
f(-x) = 8(-x)^2-8
Simplifying further:
f(-x) = 8x^2-8
Since 8x^2-8 is equal to f(x), the function is even.
In summary, the odd functions among the given options are:
- f(x) = x^3+4x
Now, let's move on to the characteristics of the parent fifth-root function f(x) = \root(5)(x):
Domain: The domain of the fifth-root function is all real numbers since we can take the fifth root of any real number.
Range: The range consists of all real numbers, as the fifth root of any real number will also be a real number.
Intervals of increase/decrease/constant: Since the fifth-root function has a positive slope for all positive values of x, it is increasing in the interval (0, +∞). The function is constant at x = 0 because the result is always 0.
Intercepts: The function intersects the x-axis at (0, 0) as the fifth root of 0 is 0.
Even/Odd/Neither: The fifth-root function f(x) = \root(5)(x) is neither even nor odd since it does not satisfy the properties of even or odd functions mentioned earlier.
Continuity/Discontinuity: The fifth-root function is continuous for all real numbers, meaning there are no breaks or jumps in the graph.
Asymptotes: The fifth-root function does not have any asymptotes. It's a smooth curve that continues indefinitely.
End behavior: As x approaches positive or negative infinity, the function becomes increasingly steep but does not approach a specific horizontal line.
I hope this explanation helps you understand both odd functions and the characteristics of the fifth-root function!
To determine which functions are odd, we need to check if f(-x) = -f(x) for all x in the domain.
The odd functions exhibit symmetry about the origin, meaning that if we reflect the graph over the y-axis, it will be identical. In other words, all points (x, y) on the graph are mirrored by (-x, -y).
Let's check the functions one by one:
1) f(x) = x/x^8 + 8x^4 - 7
To determine if it's odd, we need to check f(-x).
f(-x) = (-x)/(-x)^8 + 8(-x)^4 - 7
Simplifying, f(-x) = -x/x^8 + 8x^4 - 7 = -f(x)
Therefore, this function is odd.
2) f(x) = x^3 + 4x
f(-x) = (-x)^3 + 4(-x) = -x^3 - 4x
This is not equal to -f(x), so the function is not odd.
3) f(x) = 3x^5 + 6x^4 + 4
f(-x) = 3(-x)^5 + 6(-x)^4 + 4 = -3x^5 + 6x^4 + 4
This is not equal to -f(x), so the function is not odd.
4) f(x) = 8x^2 - 8
f(-x) = 8(-x)^2 - 8 = 8x^2 - 8
It is equal to f(x), so the function is even.
Therefore, the odd functions are:
- f(x) = x/x^8 + 8x^4 - 7
Next, let's analyze the key characteristics of the parent fifth-root function f(x) = ∜(x) = (x)^(1/5):
Domain: The function f(x) = ∜(x) is defined for all real numbers since the fifth root is defined for both positive and negative numbers.
Range: The range of f(x) includes all real numbers since the fifth root of any real number results in a real number.
Intervals where the function is increasing, decreasing, or constant: The function is increasing for positive values of x and decreasing for negative values of x. It is constant for x = 0 since the fifth root of 0 is 0.
Intercepts: The x-intercept occurs at (0, 0) since the fifth root of 0 is 0. There are no y-intercepts since the function does not intersect the y-axis.
Even or odd: The function is neither even nor odd since it does not exhibit symmetry about the y-axis or the origin.
Continuous or discontinuous: The function is continuous for all real numbers.
Asymptotes: The parent fifth-root function has no vertical asymptotes.
End behavior: As x approaches negative infinity, f(x) approaches negative infinity. As x approaches positive infinity, f(x) approaches positive infinity.
I hope this explanation helps! Let me know if you have any further questions.