Determine whether the function g(x)=x^4-7x^2 is an even, odd, or neither function. Also determine the symmetry of its graph
To determine whether a function is even, odd, or neither, we need to analyze the symmetry of its equation.
1. Even Function: A function is even if it satisfies the property g(x) = g(-x) for all x in the function's domain. In other words, if we replace x with -x in the equation, the function remains unchanged.
2. Odd Function: A function is odd if it satisfies the property g(x) = -g(-x) for all x in the function's domain. In other words, if we replace x with -x in the equation, the function changes sign.
3. Neither: If a function does not satisfy the properties of being even or odd, then it is neither an even nor an odd function.
Now, let's apply these properties to the given function g(x) = x^4 - 7x^2:
1. To test for evenness, we substitute x with -x:
g(-x) = (-x)^4 - 7(-x)^2
= x^4 - 7x^2
Since g(x) = g(-x), the function is even.
2. To test for oddness, we substitute x with -x:
-g(-x) = -((-x)^4 - 7(-x)^2)
= -(x^4 - 7x^2)
= -x^4 + 7x^2
Since -g(-x) ≠ g(x), the function is not odd.
Therefore, the function g(x) = x^4 - 7x^2 is an even function but not an odd function.
Regarding the symmetry of the graph, an even function is symmetric with respect to the y-axis. So, the graph of g(x) = x^4 - 7x^2 will be symmetric about the y-axis.