April's work in checking for the function, [tex]f(x) = x^3 - 1[/tex], to be even, odd, or neither, is incorrect. She did a mistake in step 2, where she took [tex]f(-x) = - f(x)[/tex], which was not true, since [tex]f(-x) = -x^3 - 1[/tex], whereas [tex]- f(x) = -x^3 + 1[/tex].
A function f(x) is said to be odd or even when it completes the condition:
- f(-x) = -f(x), for f(x) to be an odd function, and
- f(-x) = f(x), for f(x) to be an even function.
In the question, we are given that April was asked to determine whether [tex]f(x) = x^3 - 1[/tex], is even, odd, or neither, and are shown her work.
We are asked whether April's work is correct, and if not, what is the first step where April made a mistake.
To check for a function is even or odd, we first calculate f(-x).
April's first step was also finding f(-x), which she found accurately as [tex]f(-x) = -x^3 - 1[/tex].
Next, we need to check whether f(-x) is equal to f(x), -f(x), or neither, to check if the function is even, odd, or neither, respectively.
April, also choose the right step of checking, but did a mistake in checking, since f(-x) ≠ f(x), and also, f(-x) ≠ - f(x), since, [tex]- f(x) = -(x^3 - 1)[/tex] = [tex]-x^3 + 1[/tex], which is not equal to f(-x).
Thus, April's work in checking for the function, [tex]f(x) = x^3 - 1[/tex], to be even, odd, or neither, is incorrect. She did a mistake in step 2, where she took f(-x) = - f(x), which was not true, since [tex]f(-x) = -x^3 - 1[/tex], whereas [tex]- f(x) = -x^3 + 1[/tex].
Learn more about odd and even functions at
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