Evaluating Functions
Functions & Advanced AlgebraEvaluating a function means substituting a specific input value into the function to find the corresponding output.
Definition
Evaluating a function means plugging a specific number (or expression) into the function in place of the variable to get the output.
Example
If $f(x) = 5x - 2$, to evaluate $f(4)$: replace $x$ with $4$. $f(4) = 5(4) - 2 = 20 - 2 = 18$. The output is $18$.
Key Insight
Evaluating a function is just careful substitution. Replace every $x$ with your input value, then simplify. Order of operations matters every step.
Definition
Evaluating a function at a value $a$ means computing $f(a)$ by substituting $a$ for every occurrence of the variable in the function's rule. The input can be a number, variable, or algebraic expression.
Example
Given $f(x) = 2x^2 - 3x + 1$: $f(-2) = 2(-2)^2 - 3(-2) + 1 = 8 + 6 + 1 = 15$. $f(x + h) = 2(x+h)^2 - 3(x+h) + 1 = 2x^2 + 4xh + 2h^2 - 3x - 3h + 1$.
Key Insight
Evaluating $f(x + h)$ and simplifying $[f(x + h) - f(x)] / h$ is the foundation of the derivative in calculus, known as the difference quotient.
Definition
Evaluation is the application of the function's rule to an argument. In type theory, evaluating $f(a)$ is beta-reduction. In numerical analysis, evaluating complex functions efficiently (e.g., Horner's method for polynomials) is a key concern for computational accuracy.
Example
Horner's method evaluates $p(x) = a_n x^n + \ldots + a_0$ as $((\ldots((a_n x + a_{n-1})x + a_{n-2})x + \ldots + a_0)$, reducing $n^2$ multiplications to $n$.
Key Insight
Symbolic vs. numerical evaluation are distinct: symbolic evaluation manipulates expressions algebraically, while numerical evaluation produces floating-point approximations, each with trade-offs in precision and speed.