In mathematics, a function is an expression that returns some result based on some input, often expressed as $f(x)$. One and only one result will be returned when given an input (e.g., the function cannot be indeterminate). The result will always be the same when given the same input. Graphically, a function of $x$ will include any functions over which the line $f(x)$ has one and only one result for a given $x$. Thus, a circle, or sideways parabola, or "S" shape are not functions of $x$. ## transform a function It can be helpful to transform a function when solving equations. You can - add 0 to either side - multiply either side by 1 (or ratios with the same numerator and denominator) - add or subtract the same thing to both sides - multiply or divide both sides by the same thing - take both sides to the same exponent (e.g., square or square root both sides) - take the [[logarithm]] of both sides ## rationalize the denominator A common transformation will be rationalizing the denominator, when a root is in the denominator. To rationalize the denominator, multiply and divide by the denominator (which is a special case of 1) to pull the root into the numerator. $\frac{x}{\sqrt{x}} = \frac{x}{\sqrt{x}} * \frac{\sqrt{x}}{\sqrt{x}} = \frac{x\sqrt{x}}{x}$ ## continuity A function is continuous if you don't have to pick your pen up when graphing a function. Examples of discontinuous functions include the [[piecewise function]] the function $1/x$ because it is not defined at 0.