Questions: Determine algebraically whether the function is even, odd, or neither. √[3]7 x^2+6 even odd neither

Transcript text: Determine algebraically whether the function is even, odd, or neither. \[ \sqrt[3]{7 x^{2}+6} \] even odd neither

Solution

Solution Steps

To determine if a function is even, odd, or neither, we need to check the function's symmetry properties. A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in the domain, and it is odd if \( f(-x) = -f(x) \) for all \( x \) in the domain. If neither condition is met, the function is neither even nor odd. For the given function \( f(x) = \sqrt[3]{7x^2 + 6} \), we will calculate \( f(-x) \) and compare it to \( f(x) \) and \(-f(x)\).

Step 1: Define the Function

We are given the function \( f(x) = \sqrt[3]{7x^2 + 6} \).

Step 2: Calculate \( f(-x) \)

To determine if the function is even or odd, we calculate \( f(-x) \): \[ f(-x) = \sqrt[3]{7(-x)^2 + 6} = \sqrt[3]{7x^2 + 6} \]

Step 3: Compare \( f(-x) \) with \( f(x) \)

Now we compare \( f(-x) \) with \( f(x) \): \[ f(-x) = \sqrt[3]{7x^2 + 6} = f(x) \] Since \( f(-x) = f(x) \), the function is even.