If I were to make an equation so that the left side were a reciprocal function and the right side were an absolute value function, then the equation would be guaranteed to have at least one solution. Why?
To guarantee that the equation has at least one solution, you can set up a reciprocal function on the left side and an absolute value function on the right side of the equation. Here's an explanation of why this ensures the existence of a solution:
A reciprocal function is of the form f(x) = 1/x, where x ≠ 0. This function approaches zero as x approaches positive or negative infinity. It involves dividing by x, which can result in infinitely large or small values.
On the other hand, an absolute value function is of the form g(x) = |x|, which results in non-negative values. The function essentially takes the distance from x to zero, regardless of whether it is positive or negative. Therefore, it always yields non-negative results.
Since these two functions have different behaviors, it is reasonable to assume that their graphs intersect at least once. By setting up an equation with a reciprocal function on the left side and an absolute value function on the right side, you guarantee the existence of a solution where the two graphs intersect.
However, it is important to note that this explanation provides only a conceptual reasoning. To prove that an equation has a solution, you would need to solve the equation algebraically or graphically.