Check whether the following functions are one-one and onto f:R-R defined by f(x) =4x ^3 +5

Answer 1

One-to-one:

Your function f(x) is one-to-one if whenever you take two numbers a and b, with a not equal to b, in the domain of f then f(a) is not equal to f(b).

So to check your function, we need to check whether the equation 4a^3 + 5 = 4b^3 + 5 means we must have a=b or not.

We can simplify the equation:

1. 4a^3 + 5 = 4b^3 + 5
2. 4a^3 = 4b^3
3. a^3 = b^3

Finally, taking the cube root gives a = b and no other solution, so f i sone-to-one. (Note that it is important here that the power is 3, or any other odd number. If it was even we would have |a| = |b|, so another solution would be, for example, a = -b).

Onto:

A function f(x) is onto if for any value y you can find a value x such that f(x) = y.

For this function, we want to take the equation y = 4x^3 + and solve for x:

1. y = 4x^3 + 5
2. y - 5 = 4x^3
3. (y-5)/4 = x^3
4. x = ( (y-5)/4 )^(1/3)

For any value of y we can therefore find such an x, so f is onto. (Note that again, it is important that we have x^3; if the power on x were even then we would be taking a square root and would need (y-5)/4 to be positive, restricting the values of y we can take.)