Understanding Functions and Their Properties
A function is a relation between two variables, such that for each value of the first variable (magnitude), there corresponds one and only one value of the second. The first variable is known as the independent variable. The dependent variable is the one whose values are determined from the values of the independent variable.
Key Concepts of Functions
- The domain of a function is the set of values of X for which there is a corresponding Y value (image).
- The range of a function is the set of Y values which are the image of some value X.
- A function is increasing in an interval when, as the values of the independent variable increase, the values of the dependent variable also increase.
- A function is decreasing in an interval when, as the values of the independent variable increase, the values of the dependent variable decrease.
Rate of Change and Function Behavior
The average rate of change of a function in an interval [a, b] is the result of dividing the variation of the dependent variable, f(b) – f(a), by the increase of the independent variable, b – a.
- A function has a maximum at a point if the value of the function at this point is greater than at other nearby points.
- A function f(x) has a minimum at a point if the value of the function at this point is smaller than at other nearby points.
- If the average rate of change of a function is decreasing in an interval, the graph of the function is convex in this interval.
- If the average rate of change of a function is increasing in an interval, the graph of the function is concave in this interval.
- A function has a turning point at a point (a, f(a)) when the graph changes from convex to concave or vice-versa at this point.
Continuity and Symmetry
- A function is continuous when its graph can be drawn without lifting the pen, meaning there are no interruptions in the domain. Otherwise, it is discontinuous.
- The graph of a function is symmetric about the axis if for any value of x and its opposite -x, the corresponding image is the same.
- The graph of a function is symmetric about the origin of coordinates if for any value of x, the condition holds.
Periodicity
A function is periodic when the values of the dependent variable are repeated at equal intervals of the independent variable. In a periodic function, there is a constant T, called the period of the function, such that after an interval of length T, the graph of the function is repeated.
Types of Functions
- A constant function assigns the same value to the dependent variable for any value of the independent variable. Its algebraic expression is f(x) = k, where k is a real number. The graph is a line parallel to the x-axis.
- Linear functions describe relationships of direct proportionality between variables. They are of the form f(x) = mx, where m is a real number other than 0. The graph is a line through the origin of coordinates.
- An affine function is one where adding or subtracting a constant to the values of the dependent variable yields values proportional to the independent variable. It has the following form: f(x) = mx + b, where m and b are real numbers, and m is not 0.
The average rate of change of a linear or affine function is constant. The slope of the linear function coincides with the average rate of change.
- A quadratic function has an algebraic expression corresponding to a quadratic equation.
- When two magnitudes are directly proportional, if one doubles or triples, the corresponding value also doubles or triples.
- When two quantities are inversely proportional, if one doubles or triples, the other is halved or reduced to a third.
- When two quantities are inversely proportional, the product between corresponding values is constant, and this is called the constant of inverse proportionality.
The function of inverse proportionality transforms values of the independent variable by dividing a constant by each of these values.