Question

Determine whether each of these functions is a bijection from R to R.

1) f(x) = -3x⁴
2) f(x) = -3x²/7
3) f(x) = (x+1)/(x+2)
4) f(x) = x⁵-1

Answer 1

A bijection is a function that is both one-to-one and onto. The first three functions provided are not bijections, but the fourth function is a bijection.

Step-by-step explanation:

A function is a bijection from R to R if it is both injective (one-to-one) and surjective (onto). Let's analyze each function:

1. The function f(x) = -3x⁴ is not a bijection because it is not injective. For example, f(2) = f(-2) = 48, so it is not one-to-one.
2. The function f(x) = -3x²/7 is not a bijection because it is not surjective. The range of the function is (0,∞), so it does not cover all real numbers.
3. The function f(x) = (x+1)/(x+2) is not a bijection because it is not injective. For example, f(-3) = f(0) = -1/2, so it is not one-to-one.
4. The function f(x) = x⁵-1 is a bijection from R to R. It is both injective and surjective, covering the entire real number line.