Label each statement TRUE or FALSE.
a. The sum of two one-to-one functions is one-to-one.
b. The product of two one-to-one functions is one-to-one.
c. If f is a one-to-one function and k is a real number (constant), then the function g(x)=k*f(x)is one-to-one.
So I was thinking that a. True b. False c. True?????
To determine the correctness of each statement, let's understand the properties of one-to-one functions.
A function is said to be one-to-one (or injective) if distinct inputs always map to distinct outputs. In other words, for every pair of different elements in the domain, their corresponding outputs in the codomain should also be different.
Now, let's evaluate each statement using these properties:
a. The statement claims that the sum of two one-to-one functions is one-to-one. To verify this, consider the counterexample: Let's say we have two one-to-one functions f(x) = x and g(x) = -x. If we add them, (f(x) + g(x)), we get the constant function h(x) = 0. As h(x) = 0 is not a one-to-one function (multiple inputs map to the same output), the statement is FALSE.
b. The statement claims that the product of two one-to-one functions is one-to-one. Similar to the previous statement, consider the counterexample: Let's take f(x) = g(x) = x^2. If we multiply them, (f(x) * g(x)), we get the function h(x) = (x^2)^2 = x^4. Again, as h(x) = x^4 is not a one-to-one function (multiple inputs map to the same output), the statement is FALSE.
c. The statement claims that if f is a one-to-one function and k is a real number (constant), then the function g(x) = k * f(x) is one-to-one. To validate this, we need to show that distinct inputs result in distinct outputs. Let's assume that f(a) = f(b) for some distinct inputs a and b. In that case, k * f(a) = k * f(b). Since k is a constant and f(a) = f(b), both sides of the equation are equal. Therefore, the statement is TRUE.
In summary:
a. FALSE
b. FALSE
c. TRUE