A function is a fundamental concept in mathematics. It's a special type of relation between two sets where each input has exactly one output. Think of it like a machine that takes an input and produces a unique output. The domain is the set of all possible inputs, the codomain contains all possible outputs, and the range is the set of actual outputs the function produces. This one-to-one correspondence is what makes a relation a function.
Function notation uses f(x) to represent a function, read as 'f of x'. This means we apply function f to input x. Let's look at examples: f(x) equals 2x plus 1 is a linear function, g(x) equals x squared is a quadratic function, and h(x) equals 1 over x is a rational function. To evaluate a function, we substitute the input value. For instance, f(3) equals 2 times 3 plus 1, which equals 7. Similarly, g(2) equals 2 squared, which equals 4. These functions can be visualized as graphs on a coordinate plane.
Domain and range are crucial concepts for understanding functions. The domain is the set of all possible input values, while the range is the set of all actual output values. For a linear function like f(x) equals 2x plus 1, both domain and range include all real numbers. However, for rational functions like g(x) equals 1 over x, we have restrictions. The domain excludes zero because division by zero is undefined, and the range also excludes zero because the function never outputs zero. These restrictions are visualized on graphs using color-coded arrows and annotations.
Functions can be classified into different types based on their properties. A one-to-one or injective function means each output corresponds to a unique input. An onto or surjective function uses every possible output value. A bijective function is both one-to-one and onto. Functions also have behavioral properties: increasing functions rise from left to right, decreasing functions fall, even functions are symmetric about the y-axis, and odd functions have rotational symmetry about the origin. These classifications help us understand and analyze function behavior.
Functions are everywhere in real life. Temperature conversion from Celsius to Fahrenheit follows the linear function F equals nine-fifths C plus 32. Compound interest uses exponential functions where amount equals principal times one plus rate to the power of time. Distance-time relationships in physics use linear functions like distance equals velocity times time. Population growth models use exponential functions. These mathematical models help us understand, predict, and solve real-world problems in science, economics, engineering, and daily life.