Evaluating Functions — Definition, Formula & Examples
Evaluating a function means replacing the variable with a given input value and calculating the result. If you're told and asked to find , you substitute 5 for and simplify.
To evaluate a function at a value is to compute by substituting for every occurrence of the independent variable in the function's rule, then performing all indicated operations to obtain a single output value.
How It Works
Identify the input value (the number or expression inside the parentheses). Replace every instance of the variable in the function rule with that input — use parentheses around the substituted value to avoid sign errors. Simplify using order of operations (exponents first, then multiplication/division, then addition/subtraction). The result is the function's output, or the value of the dependent variable at that input.
Worked Example
Problem: Given f(x) = 3x² − 4x + 1, find f(−2).
Substitute: Replace every x with (−2).
Apply exponent: Compute (−2)² = 4, then multiply by 3.
Simplify: Compute −4(−2) = 8, then add all terms.
Answer: f(−2) = 21
Why It Matters
Evaluating functions is the foundation for graphing, solving equations, and analyzing real-world models in Algebra 1 and beyond. Every time you plug a value into a formula — from physics equations to financial models — you are evaluating a function.
Common Mistakes
Mistake: Forgetting parentheses when substituting a negative number, turning (−2)² into −2² and getting −4 instead of 4.
Correction: Always wrap the substituted value in parentheses. Write 3(−2)², not 3·−2², so the exponent applies to the entire input including its sign.