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.
To determine whether each statement is true or false, we need to understand the concept of one-to-one functions.
A one-to-one function, also known as an injection, is a function in which each element of the domain is mapped to a unique element in the range. This means that the function does not map two different elements from the domain to the same element in the range.
Now let's evaluate each statement and determine if it is true or false:
a. "The sum of two one-to-one functions is one-to-one."
To determine the truth of this statement, we should look for counterexamples. Let's consider two functions, f(x) = x and g(x) = -x. Both of these functions are one-to-one since they map each input to a unique output. However, when we add them together, h(x) = f(x) + g(x) = x + (-x) = 0, we can see that the output (0) is not unique for different inputs. For example, h(1) = 0 and h(-1) = 0. Therefore, this statement is FALSE.
b. "The product of two one-to-one functions is one-to-one."
Again, we should search for counterexamples to test the validity of this statement. Let's consider two functions, f(x) = x and g(x) = 1/x. Both of these functions are one-to-one since they map each input to a unique output. However, when we multiply them together, h(x) = f(x) * g(x) = x * (1/x) = 1, we can see that the output (1) is the same for any input except for x = 0. For example, h(2) = h(3) = 1. Therefore, this statement is FALSE.
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."
To determine the truth of this statement, let's analyze it logically. Since f is a one-to-one function, it means that each input x is mapped to a unique output f(x). Now, if we multiply the range of f(x) by a constant k, it does not affect the uniqueness of the mapping. In other words, multiplying the output by a constant does not introduce any collisions or repetitions. Thus, each input x in g(x) = k * f(x) is still uniquely mapped to a unique output k * f(x). Therefore, this statement is TRUE.
To summarize:
a. FALSE
b. FALSE
c. TRUE