If I were to make an equation so that the left side were reciprocal function and the right side were an absolute value function then the equation would be guaranteed to have at least one solution. Why?
Why not follow the same steps I showed you for the questions similar to this one.
Let me know how you make out.
Im very confused about this one too, can someone help?
To understand why an equation with a reciprocal function on the left side and an absolute value function on the right side is guaranteed to have at least one solution, let's analyze the properties of these functions.
First, let's consider the reciprocal function, which is of the form f(x) = 1/x. The important characteristic of this function is that it never equals zero, except at x = 0. This means that the reciprocal function approaches infinity as x approaches 0 from the positive side, and approaches negative infinity as x approaches 0 from the negative side.
On the other hand, the absolute value function, denoted as |x|, always returns a non-negative value. If the input is positive or zero, the output is equal to the input. If the input is negative, the output becomes positive by taking the absolute value.
Now, when you set up an equation with a reciprocal function on the left side and an absolute value function on the right side, let's say f(x) = g(x), where f(x) represents the reciprocal function and g(x) represents the absolute value function.
Since the reciprocal function never equals zero, we can deduce that g(x) (the absolute value function) must also be non-zero. This is because if g(x) were equal to zero, it would imply that f(x) (the reciprocal function) is equal to zero, which is not possible.
Therefore, we have a situation where a non-zero value on the right side of the equation is equal to a function that is always non-zero on the left side. In other words, the equation f(x) = g(x) will always have at least one solution.
However, it's important to note that this analysis only guarantees the existence of a solution; it does not provide the methodology to find the specific solution(s).