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.
a. TRUE
b. TRUE
c. TRUE
To determine whether each statement is true or false, we need to understand the properties and definitions of one-to-one functions.
A function is considered one-to-one (also known as injective) if each element in the domain corresponds to a unique element in the codomain. In other words, no two elements in the domain can have the same output in the codomain.
Let's evaluate each statement:
a. The sum of two one-to-one functions is one-to-one.
To determine if this statement is true or false, we can counterexample it. Consider two one-to-one functions: f(x) = x and g(x) = -x. If we take their sum, h(x) = f(x) + g(x) = x + (-x) = 0, we can see that multiple elements in the domain (any x) map to the same output (0 in this case). Therefore, the sum of two one-to-one functions is not necessarily one-to-one.
Therefore, statement a is FALSE.
b. The product of two one-to-one functions is one-to-one.
Similar to statement a, we can counterexample this statement. Consider two one-to-one functions: f(x) = x and g(x) = 1/x. If we multiply them, h(x) = f(x) * g(x) = x * (1/x) = 1, we can see that all elements in the domain (except for x = 0) map to the same output (1 in this case). Hence, the product of two one-to-one functions is not necessarily one-to-one.
Therefore, statement b is also 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 prove or disprove this statement, we need to consider the properties of one-to-one functions under scalar multiplication.
When we multiply a one-to-one function f(x) by a constant k, the resulting function g(x) = k * f(x) will still be one-to-one as long as k is not equal to zero. That's because scalar multiplication does not change the uniqueness of the mapping from domain to codomain. Every element in the domain still corresponds to a unique element in the codomain.
However, if k is equal to zero, then the resulting function g(x) = 0 * f(x) = 0 will not be one-to-one because all elements in the domain will map to the same output, which is 0.
Therefore, statement c is TRUE, as long as k is not equal to zero.
Summary:
a. FALSE
b. FALSE
c. TRUE
a) - yes
the sum of ax+b and cx+d is still linear
b) - no
the product of ax+b with cx+d produces a term in x^2. Quadratics are not one-to-one
c) - yes
multiplying a linear like ax+b by a constants keeps it linear.