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An implicit function is one where the relationship between the dependent variable and independent variable is not expressed directly. Unlike explicit functions, where you can write the dependent variable (often y) solely in terms of the independent variable (often x), implicit functions involve both variables mixed together. The given exercise \(x^{2}+y^{2}=9\) is a classic example of an implicit relationship because neither variable is isolated on one side of the equation initially.

While working with implicit functions, our goal is often to express one variable in terms of the other. In the provided solution, the steps guide us through altering the implicit equation into two explicit functions, representing the positive and negative square roots, respectively. Understanding how to work with implicit equations is crucial as they commonly appear in various fields of mathematics and science.

The process of isolating variables is a fundamental technique in algebra. It’s the method of rearranging an equation to make one variable the subject of the formula. The steps provided demonstrate isolating the variable y by moving all other terms to the other side of the equation. It’s crucial to perform the same operation on both sides to maintain equality. After rearranging, we apply the square root to both sides because y is squared, and we want to solve for y.

• Move terms that do not contain the variable of interest to the other side of the equation
• Use inverse operations to undo addition, subtraction, multiplication, or division
• Remember to consider all possible solutions, like the positive and negative square roots in this context

Isolating variables is not only important in solving equations but also in understanding how changing one variable affects another, which is a key concept in functions.

The square root function is a special type of function that is crucial in solving quadratic equations like the one in our exercise. Given a number x, the square root function outputs a number y, such that \(y^2 = x\). It is important to note that every positive number x has two square roots: one positive and one negative (except for zero, which has a single square root of zero).

In the solution steps, applying the square root to both sides of the equation \(y^{2}=9-x^{2}\) results in \(y=\sqrt{9-x^{2}}\) and \(y=-\sqrt{9-x^{2}}\). These represent the positive and negative square root functions of the expression \(9-x^2\). This dual nature leads us to define two separate functions, f(x) and g(x), to capture all possible y-values that satisfy the original equation.