function value — from formula anatomyIn mathematics and physics, a function value is the output of a function when a specific input is given. For a function f(x), the value at x = a is written f(a). It is the fundamental building block of formulas: every equation that predicts a quantity—from the trajectory of a projectile to the energy of a quantum state—is ultimately a rule for computing function values.
The concept of a function evolved over centuries, with formal definitions emerging in the 19th century through mathematicians like Dirichlet and Riemann. The notation f(x) was popularized by Euler in the 18th century.
A function is a rule that assigns exactly one output to each input. To find the value, you substitute the input into the rule and simplify. For example, if f(x) = x² + 1, then f(3) = 10. In physics, function values often depend on variables like time, position, or temperature, and they encode how a system behaves under different conditions.
Function values are used everywhere in physics and engineering—from calculating the force on a charged particle to predicting the pressure in a gas at a given volume and temperature.
The word 'function' was introduced by Leibniz in 1694, but it originally meant any quantity related to a curve, not the modern input-output rule.
| f(x_i) | variable | model prediction at input x_i — function value |
| y_i | variable | observed data point — data value |
| σ_i | variable | standard deviation of measurement error — standard deviation |
| ∏ | operator | Product over all data points i — product operator |
| exp | operator | Exponential function — exponential function |
| θ* | variable | Fixed point value — fixed point |
| F(θ*) | variable | Function evaluated at fixed point — function value |
| f''(x) | variable | Second derivative of function f at x — second derivative per metre squared |
| f(x+h) | variable | Function value at x+h — function value |
| f(x-h) | variable | Function value at x-h — function value |
| f(x) | variable | Function value at x — function value |
| h | variable | Step size — spacing metre |
| \approx | operator | Approximately equal to — approximation |
| f | variable | scalar function value — function value |
| x | variable | input vector — vector argument |
| Q | variable | quadratic coefficient matrix — quadratic coefficient matrix |
| b | variable | linear coefficient vector — linear coefficient vector |
| c | variable | constant term — constant term |
| T | operator | flips a matrix or vector over its diagonal — transpose operator |