Evaluating Functions
Understand function notation f(x), evaluate functions at given inputs, interpret domain and range, and see how functions model real-world relationships.
Every time you use a formula — converting Celsius to Fahrenheit, calculating compound interest, predicting population growth — you're using a function. In programming, every method or procedure is a function. In science, every law (F = ma, E = mc²) is a function. The function is the universal language of quantitative relationships, and f(x) notation is how mathematicians, scientists, and engineers write them.
- Read and write function notation: f(x), g(t), h(n)
- Evaluate a function at any given input value
- Identify the domain (valid inputs) and range (resulting outputs) of a function
- Use functions to model real-world relationships
\(f(x) = 3x^2 - 2x + 1\) defines a function named f with input x. The notation \(f(4)\) means "evaluate f when x = 4": substitute 4 everywhere you see x.
\[ f(4) = 3(4)^2 - 2(4) + 1 = 3(16) - 8 + 1 = 48 - 8 + 1 = 41 \]Think of a function as a machine: you put in an input (x), it applies the rule, and out comes one output (f(x)). Same input → same output, every time.
Function Machine — f(x) = 2x + 5
x = 3
apply the rule
f(3) = 11
2(3) + 5 = 6 + 5 = 11
Given \(f(x) = 5x - 3\), find \(f(2)\) and \(f(-1)\).
- \(f(2) = 5(2) - 3 = 10 - 3 = 7\)
- \(f(-1) = 5(-1) - 3 = -5 - 3 = -8\)
Each input gives exactly one output — that's what defines a function.
Given \(g(x) = x^2 + 1\), find \(g(a+1)\).
- Replace x with (a+1): \(g(a+1) = (a+1)^2 + 1\)
- Expand: \((a+1)^2 = a^2 + 2a + 1\)
- Add 1: \(g(a+1) = a^2 + 2a + 2\)
The temperature in a city during a cold snap is modeled by \(T(h) = -0.5h + 60\) where h = hours after midnight and T = temperature in °F.
- \(T(0)\) = midnight temperature: \(-0.5(0) + 60 = 60°F\)
- \(T(12)\) = noon temperature: \(-0.5(12) + 60 = -6 + 60 = 54°F\)
- \(T(h) = 32\) (freezing): \(-0.5h + 60 = 32\) → \(h = 56\) → 56 hours after midnight
The function predicts temperature at any hour — and we can solve backward to find when freezing occurs.
Given \(f(x) = 2x^2 - x + 3\):
- Find \(f(0)\)
- Find \(f(-2)\)
- For what value of x does \(f(x) = 12\)?
▶ Show Answers
- \(f(0) = 0 - 0 + 3 =\) 3
- \(f(-2) = 2(4) - (-2) + 3 = 8 + 2 + 3 =\) 13
- \(2x^2 - x + 3 = 12\) → \(2x^2 - x - 9 = 0\) → x ≈ 2.34 or −1.84 (two solutions)
- Reading f(x) as multiplication: \(f(3)\) does NOT mean f × 3. It means "evaluate the function f at the input x = 3."
- Forgetting to square the entire substituted expression: For \(f(x) = x^2\), \(f(a+1) = (a+1)^2\), NOT \(a^2 + 1\).
- Confusing domain and range: Domain = inputs (x values); Range = outputs (y or f(x) values). Check which is being asked.
- f(x) = rule: function notation packages a rule so we can evaluate it at any input.
- Evaluate by substituting the input for every x — use parentheses to avoid sign errors.
- Domain = valid inputs; range = resulting outputs.
- Functions model the world — every formula with a variable input is a function.
In programming, functions are the cornerstone of all code: you define a function with inputs (parameters) and a rule (the body), then call it at any input. The mathematical definition of a function — one output per input, encapsulated in a named rule — is identical to a programming function. Every software developer uses function thinking dozens of times per day.
Calculator Connection
Use the Function Evaluation Calculator to evaluate any function at a given input with step-by-step substitution shown. The Function Plotter graphs your function so you can visualize the domain, range, and behavior across all inputs.
Try it with the Calculator
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Evaluating Functions - Quiz
5 questions per attempt · Intermediate · Pass at 70%
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