Question

What is the standard form of the rational function r(x)=mx b/cx d, where c=0? a) r(x) = a(1/x-h) + k b) r(x) = a(1/x-h) - k c) r(x) = a(1/x+h) + k d) r(x) = a(1/x+h) - k

Answer 1

The standard form of the rational function r(x) = (mx + b)/(cx + d) with c = 0 simplifies to a linear equation that resembles y = mx + b, which means none of the given options are correct, as they are all forms of a transformed reciprocal function rather than a rational function.

Step-by-step explanation:

The standard form of the rational function
`r(x) = (mx + b)/(cx + d)` where c = 0 is not one of the options provided because with c being zero, the given function simplifies to
`r(x) = (mx + b)/(d)` which is a linear function and not a rational function anymore. In its simplified form, it represents a straight line equation similar to the standard line equation y = mx + b. Since the denominator is just a constant d and not dependent on x, there is no vertical asymptote that would be represented in the forms shown in the options, which involve variations of a(1/(x-h)). If we interpret the question as having a typo and c should not be zero, then none of the answer choices would necessarily represent the standard form of a rational function either. They are transformations of the reciprocal function rather than a standard rational function involving linear expressions in both numerator and denominator.

Answer 2

The rational function r(x)=mx b/cx d is undefined when c=0, because division by zero is an undefined operation in mathematics. None of the given options can be valid because they all have a variable in the denominator.

The question is asking for the standard form of the rational function r(x)=mx b/cx d, where c=0. In mathematics, when we have a rational function and the denominator is zero, the function is undefined. This is because we cannot divide by zero in mathematics; it's an undefined operation.

However, if you are trying to simplify the expression to a standard form given that the denominator part is zero, it's also not possible because, as previously mentioned, any fraction with the denominator being zero is undefined. With that said, none of the options a), b), c), and d) can be accurate in this situation, since all have a variable in the denominator. It indicates that you either made a mistake or there's a typo in your problem.