Let g(x)=3x^2+14x-15. Determine if g is one to one or onto.
A. g:N->N
B. g:R->R
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To determine if the function g(x) = 3x^2 + 14x - 15 is one-to-one or onto, let's understand the definitions of these terms:
- One-to-one (Injective): A function is one-to-one if each element of the domain corresponds to a unique element in the range. In other words, if two different elements in the domain map to the same element in the range, the function is not one-to-one.
- Onto (Surjective): A function is onto if every element in the range has a corresponding element in the domain. In other words, for every element in the range, there is at least one element in the domain that maps to it.
Now, let's check if g(x) is one-to-one and onto for each case:
A. g: N -> N (natural numbers):
Since the function maps from the natural numbers to the natural numbers, we need to check if it is one-to-one and onto in this context.
To determine if g(x) is one-to-one, we can check if it passes the horizontal line test. However, since the domain and range are both infinite, we cannot test every point. Instead, we can analyze the function's derivative.
Taking the derivative of g(x) gives g'(x) = 6x + 14.
Since the derivative is always positive (unless it is zero), g(x) is an increasing function. This means that for any two different values of x in the domain, g(x) will have a different value. Therefore, g(x) is one-to-one.
To determine if g(x) is onto, we need to see if every natural number can be achieved as a result of plugging in a value from the natural numbers into g(x).
Let's check if we can find a value of x such that g(x) gives a specific natural number, for example, 1:
3x^2 + 14x - 15 = 1
Rearranging the equation:
3x^2 + 14x - 16 = 0
Solving this quadratic equation will give us the values for x. However, upon solving, we find that there are no natural number solutions to this equation.
Therefore, g(x) is not onto for g: N -> N.
B. g: R -> R (real numbers):
Similarly, we will analyze if g(x) is one-to-one and onto for the function mapping from real numbers to real numbers.
To determine if g(x) is one-to-one, we need to check the derivative of g(x), which we already found as g'(x) = 6x + 14.
Since the derivative is always positive (unless it is zero), g(x) is an increasing function, which implies that it is one-to-one.
To determine if g(x) is onto, we need to check if for any real number y, we can find an x such that g(x) = y.
We can rewrite the function as:
3x^2 + 14x - 15 - y = 0
This is a quadratic equation, and we need to check if it has a real solution for any value of y. Since the discriminant (b^2 - 4ac) is positive, there are always real solutions to the quadratic equation.
Therefore, g(x) is onto for g: R -> R.
In summary:
A. g: N -> N is not one-to-one and not onto.
B. g: R -> R is one-to-one and onto.