Evaluating functions
Evaluating a function means substituting a specific input for the function's variable and simplifying to find the output. For f(x) = 2x + 3, evaluating f(4) means replacing x with 4 to get f(4) = 11.
Evaluating a function means finding its output for a specific input. A function like f(x) = 2x + 3 is a rule: whatever value goes in for x, the rule tells you what comes out. To evaluate f(4), replace every x with 4 and simplify: f(4) = 2(4) + 3 = 11.
The substitution rule applies no matter what the input looks like. Inputs can be negative numbers or fractions — f(−2) = 2(−2) + 3 = −1 — but they can also be expressions. Evaluating f(a + 1) means replacing every x with (a + 1): f(a + 1) = 2(a + 1) + 3 = 2a + 5. Keeping parentheses around the substituted input is the single best habit for avoiding sign and distribution errors, especially with squared terms: if g(x) = x², then g(−3) = (−3)² = 9, not −9.
Function evaluation also extends to composed functions, where one function's output becomes another's input. To find f(g(2)), evaluate the inner function first, then feed the result into the outer one. Tests also present functions as tables and graphs — evaluating from a graph means finding the y-value at the given x.
Evaluating functions is a foundational skill on the SAT math section and the Praxis Core math exam, and it underlies nearly every algebra topic that follows, from exponential functions to domain and range. Expect straightforward substitution questions as well as evaluations embedded in multi-step problems.
Key takeaways
- To evaluate a function, substitute the given input for every occurrence of the variable, then simplify.
- Always wrap the substituted input in parentheses to avoid sign errors, e.g. g(−3) = (−3)² = 9.
- Inputs can be numbers or whole expressions, as in f(a + 1).
- For composite functions like f(g(2)), evaluate the inner function first.