34. Determine whether the function \( f \) is even, odd or neither. \( \begin{array}{ll}\text { (i) } f(x)=1-x^{4} & \text { (ii) } f(x)=x^{2}-x \\ \text { (iii) } f(x)=3 x^{3}+2 x-1 & \text { (iv) } f(x)=x+\frac{1}{\text { (i) }}\end{array} \)

We wish to decide if each given function f is even, odd, or neither. Recall: • A function is even if f(–x) = f(x) for all x in its domain. • A function is odd if f(–x) = –f(x) for all x in its domain. • Otherwise, the function is neither even nor odd. Below we examine each function. ────────────────────────────── (i) Let f(x) = 1 – x⁴. Calculate f(–x):   f(–x) = 1 – (–x)⁴ = 1 – x⁴   (since (–x)⁴ = x⁴). Since f(–x) = f(x), the function is even. ────────────────────────────── (ii) Let f(x) = x² – x. Calculate f(–x):   f(–x) = (–x)² – (–x) = x² + x. Compare f(–x) with f(x):   f(x) = x² – x  and  f(–x) = x² + x. They are not identical, and also note that f(–x) ≠ –[x² – x] = –x² + x. Thus, f is neither even nor odd. ────────────────────────────── (iii) Let f(x) = 3x³ + 2x – 1. Calculate f(–x):   f(–x) = 3(–x)³ + 2(–x) – 1 = –3x³ – 2x – 1. Now, consider –f(x):   –f(x) = –(3x³ + 2x – 1) = –3x³ – 2x + 1. The constant term does not match (–1 versus +1), so f(–x) ≠ –f(x). Also, f(–x) ≠ f(x) because the signs of the x³ and x terms change. Thus, f is neither even nor odd. ────────────────────────────── (iv) The function appears as   f(x) = x + 1/(x) which is a common example. (Note: Sometimes notation might seem confusing. Here we interpret it as f(x)= x + (1/x).) Calculate f(–x):   f(–x) = (–x) + 1/(–x) = –x – 1/x. This can be factored as:   f(–x) = –(x + 1/x) = –f(x). Thus, f is odd (keeping in mind that the domain excludes x = 0). ────────────────────────────── Summary of Answers:   (i) Even.   (ii) Neither.   (iii) Neither.   (iv) Odd.