Question

Prove that whether y=X^2 -3 is a one-one function or not.

Answer 1

Explanation:

To determine whether y = x^2 - 3 is a one-to-one function, we need to use the horizontal line test. The horizontal line test states that a function is one-to-one if and only if every horizontal line intersects the graph of the function at most once. Here are the steps to show whether y = x^2 - 3 is a one-to-one function:

Step 1: Assume that y = x^2 - 3 is not a one-to-one function.

Step 2: To disprove this assumption, we need to find two distinct values of x that map to the same value of y.

Let y = x^2 - 3

Suppose x1 and x2 are two distinct values such that y(x1) = y(x2)

Then we have x1^2 - 3 = x2^2 - 3

Rearranging, we get x1^2 = x2^2

Taking square root of both sides, we get |x1| = |x2| since x1 and x2 are distinct values.

Therefore, we have two cases:

Case 1: x1 = -x2

Substituting this into the original equation y = x^2 - 3, we get y(x1) = y(-x1)

This means that both x1 and -x1 map to the same value of y. Thus, y = x^2 - 3 is not a one-to-one function.

Case 2: x1 ≠ -x2

In this case, we have found two distinct values of x that map to the same value of y, which means that y = x^2 - 3 is not a one-to-one function.

Step 3: Since we have found two distinct values of x that map to the same value of y, we have proven that y = x^2 - 3 is not a one-to-one function.

Therefore, we can conclude that the function y = x^2 - 3 is not one-to-one.

Answer 2

To determine whether the function f(x) = x^2 - 3 is one-to-one or not, we need to check whether different values of x in the domain of the function produce different values of y.

Assume that there exist two different values x1 and x2 such that f(x1) = f(x2). Then, we have:

x1^2 - 3 = x2^2 - 3

Adding 3 to both sides, we get:

x1^2 = x2^2

Taking the square root of both sides, we get:

|x1| = |x2|

Therefore, we can see that there are two possible cases:

If x1 = x2, then f(x1) = f(x2), which means the function is not one-to-one.

If x1 ≠ x2 but |x1| = |x2|, then f(x1) = f(x2), which means the function is not one-to-one.

Therefore, we can conclude that the function f(x) = x^2 - 3 is not one-to-one. This is because different values of x can produce the same value of y. For example, both f(2) and f(-2) give the same value of 1. Therefore, the function is not invertible, and its inverse cannot be uniquely defined.