--- title: "Function notation and evaluating functions" description: "Read f(x) as a rule that turns each input into one output, evaluate a function at given values, and solve f(x) = k." canonical: https://lightmysky.com/learn/mathematics/function-notation-and-evaluating-functions-mt_YWKWzHK_6Z source: https://lightmysky.com/learn/mathematics/function-notation-and-evaluating-functions-mt_YWKWzHK_6Z.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Function notation and evaluating functions Read f(x) as a rule that turns each input into one output, evaluate a function at given values, and solve f(x) = k. Subject: Mathematics · Area: Algebra · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/function-notation-and-evaluating-functions-mt_YWKWzHK_6Z ## Ready when they can - Work out f(3) and f(-2) for a given rule - Solve f(x) = 0 for a quadratic rule - Explain why each input is allowed only one output ## Lesson: Function notation and evaluating functions An nth-term rule takes a position number as input and gives a term as output. A function also gives exactly one output for each allowed input. We name a function f and write its rule, for example f(x) = 3x - 2. The x in f(x) is a placeholder for the input. Writing f(4) means replace x with 4 in the rule and calculate the output. *(drawing: For each allowed input, the function gives exactly one output.)* Evaluate by replacing every x with the input. For f(x) = 3x - 2, f(-1) = 3(-1) - 2 = -5. Here 3(-1) means multiplication, not 3 - 1. Different inputs may share an output: 3 and -3 both give 5 under x² - 4. Each allowed input still has exactly one output. To solve f(x) = k, find the input or inputs that give output k. Set the rule equal to k and solve the equation. When k = 0, these inputs are the roots. A linear rule with nonzero slope and all real inputs gives one input for every real output. A quadratic may give two real inputs, one at its turning point, or none when k is outside its range. **Example.** For f(x) = 3x - 2, evaluate three inputs. f(4) = 3(4) - 2 = 10. f(-1) = 3(-1) - 2 = -5. f(0) = 3(0) - 2 = -2. Keep negative inputs in brackets. In a polynomial rule, substituting zero removes every term containing x, leaving the constant term. **Example.** For g(x) = x² - 4, g(3) = 9 - 4 = 5 and g(-3) = (-3)² - 4 = 9 - 4 = 5. The whole negative input is squared: (-3)(-3) = 9. Writing -9 - 4 would square the magnitude and then apply a minus sign, which is not this substitution. Also g(2) = 4 - 4 = 0. Opposite inputs may share an output without breaking the one-output rule. **Example.** For h(x) = x² - 5x + 6, find inputs giving 0: x² - 5x + 6 = 0 factors as (x - 2)(x - 3) = 0. The inputs are 2 and 3. Check h(2) = 4 - 10 + 6 = 0. For output 2, subtract 2 from the rule: x² - 5x + 4 = 0. This factors as (x - 1)(x - 4) = 0, giving inputs 1 and 4. In each case, set the rule equal to the requested output. An input may be an expression. For p(t) = 4t - 3, find p(x + 2) by replacing every t with the whole expression x + 2: 4(x + 2) - 3 = 4x + 5. Keep brackets during substitution so the coefficient multiplies every term in the input. To recover p(x) = ax + b from two values, turn each value into an equation. If p(2) = 5 and p(3) = 8, then 2a + b = 5 and 3a + b = 8. Subtract to get a = 3, then use 2a + b = 5 to find b = -1. Now p(5) = 3(5) - 1 = 14. Count distinct inputs when solving a quadratic rule. The equation (x - 7)² = 0 has only input 7, since both factors give that same value. A square is nonnegative for real inputs. Thus (x + 6)² + 2 is always at least 2 and never gives output 0. Expand a squared bracket to check whether a quadratic can be written this way. **Recap.** Read f(x) as a rule with a placeholder, evaluate by substituting the input with care for negative numbers, and solve f(x) = k by setting the rule equal to k and solving for the inputs. ## Practice 28 questions on this page, each with its working shown. ## Needs first - [Solving quadratic equations by factorising](https://lightmysky.com/learn/mathematics/solving-quadratic-equations-by-factorising-mt_tg_sqRzDjU) ## Opens up - [Modulus Functions and Modulus Equations](https://lightmysky.com/learn/mathematics/modulus-functions-and-modulus-equations-mt__W-aDpT5jz) - [Transformations of Graphs](https://lightmysky.com/learn/mathematics/transformations-of-graphs-mt_691_c-7Z5M) - [Domain and Range of a Function](https://lightmysky.com/learn/mathematics/domain-and-range-of-a-function-mt_b_4zoHz8Jc) - [Composite functions](https://lightmysky.com/learn/mathematics/composite-functions-mt_jqGJHXgYNw) - [Linear Maps and Their Matrices](https://lightmysky.com/learn/mathematics/linear-maps-and-their-matrices-mt_pzrfiQFdto)