Determine algebraically whether the function is even, odd, or neither even nor odd.
f as a function of x is equal to 14 times the cube root of x
All even functions satisfy:
f(x)=f(-x)
All odd functions satisfy:
f(-x)=-f(x)
If a function does not satisfy either one of the above, it is neither odd nor even.
Would you try to write the following expression in mathematical form?
"f as a function of x is equal to 14 times the cube root of x"
To determine whether the function f(x) = 14 * ∛x is even, odd, or neither even nor odd, we need to evaluate whether it satisfies the properties of even and odd functions.
1. Even function: A function f(x) is even if f(-x) = f(x) for all x in the domain of f.
Let's check if f(-x) = f(x):
f(-x) = 14 * ∛(-x) = 14 * (-∛x) = -14 * ∛x
Since f(-x) = -14 * ∛x ≠ f(x) = 14 * ∛x, the function is not even.
2. Odd function: A function f(x) is odd if f(-x) = -f(x) for all x in the domain of f.
Let's check if f(-x) = -f(x):
f(-x) = 14 * ∛(-x) = 14 * (-∛x) = -14 * ∛x
Since f(-x) = -14 * ∛x = -f(x) = -14 * ∛x, the function satisfies the condition for odd functions.
Therefore, the function f(x) = 14 * ∛x is an odd function.
To determine whether a function is even, odd, or neither, we need to analyze its symmetry properties.
A function is even if it satisfies the condition f(x) = f(-x) for all values of x in its domain. Geometrically, even functions are symmetric with respect to the y-axis.
A function is odd if it satisfies the condition f(x) = -f(-x) for all values of x in its domain. Geometrically, odd functions are symmetric with respect to the origin.
Let's analyze the given function f(x) = 14 * cuberoot(x) algebraically:
1. Checking for evenness:
f(x) = 14 * cuberoot(x)
f(-x) = 14 * cuberoot(-x) = 14 * -cuberoot(x)
Since f(x) is not equal to f(-x), we can conclude that the function is not even.
2. Checking for oddness:
f(x) = 14 * cuberoot(x)
-f(-x) = -14 * cuberoot(-x) = -14 * -cuberoot(x) = 14 * cuberoot(x)
Since f(x) is equal to -f(-x), we can conclude that the function is odd.
Therefore, the given function f(x) = 14 * cuberoot(x) is odd.