Determine whether f(x)=x^2 is one to one.
I have the answer as yes. Is this right
I don't know what you mean by "one to one" for a function. It is true that for each x there is one y.
for every value of x there is one and only one value of f(x)
[That means f(x) is a function of x]
however for every value of f(x) there are two values of x
[ that means that the inverse is not a function]
Not a function but is it one to one?
No it is not
It sure looks obvious to me that was what Damon told you.
To have a one-to-one, both the relation and its inverse must be functions.
Yours is not
Thanks
To determine whether the function f(x) = x^2 is one-to-one, we need to analyze its behavior for different values of x.
A function is said to be one-to-one (or injective) if every distinct input value (x) produces a distinct output value (f(x)). In other words, if two different input values produce the same output value, the function is not one-to-one.
To check this for f(x) = x^2, we can consider the following:
1. Assume two different input values, say x1 and x2, produce the same output value: f(x1) = f(x2).
- This means x1^2 = x2^2.
2. To find if there's a contradiction or not, we can take the square root of both sides:
- √(x1^2) = √(x2^2).
- |x1| = |x2|, where | | represents the absolute value.
If we consider two cases:
- If x1 = x2, then the equation |x1| = |x2| holds true, but this does not contradict that the function is one-to-one.
- If x1 ≠ x2, then the equation |x1| = |x2| contradicts the assumption that f(x) = x^2 is one-to-one. This occurs because taking the square root of both sides removes the sign, leading to different input values producing the same output, violating the one-to-one property.
Based on this analysis, we conclude that f(x) = x^2 is not one-to-one since there exist different input values that produce the same output value.