What is a function?

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A function is a rule that assigns exactly one output to each input. Every input (x-value) maps to one and only one output (y-value). Example: f maps every temperature in °C to a temperature in °F — one input, one output.

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First mapping (1→5, 2→7, 3→5): YES, this is a function. Two inputs (1 and 3) share the same output — that is allowed. Second mapping (1→5, 1→9): NOT a function. Input 1 maps to two different outputs — that breaks the rule.

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Example: "Country → Capital city." Each country has exactly one capital — every input (country) maps to exactly one output (capital). Non-example: "Person → Friend" — a person can have many friends, so one input maps to many outputs.

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Yes — this is perfectly fine and does NOT stop something from being a function. What is NOT allowed: one input mapping to two different outputs. Example: f(2) = 5 and f(3) = 5 is fine. But f(2) = 5 and f(2) = 9 means it is not a function.

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f(x) is the output of the function f when the input is x. Read it as "f of x." f is the name of the function. x is the input. f(x) is the corresponding output. Example: if f(x) = 2x + 1, then f(3) = 7.

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f(x) = 4x − 3. Replace y with f(x). The name "f" is conventional but any letter works (g, h, p, etc.). Both y = 4x − 3 and f(x) = 4x − 3 describe the same rule.

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g(t): apply the same rule but with t as the input → g(t) = t² + 1. g(a + 1): replace every x with (a + 1) → g(a + 1) = (a + 1)² + 1. The letter inside the bracket is always the input — substitute it everywhere x appears.

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f(x) is not multiplication — the parentheses here mean "function of," not "times." f(x) = 4x + 2 does not mean f × x = 4x + 2. f is the function name; f(x) is the output value when the input is x.

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Substitute a for every x in the function rule, then simplify. Example: f(x) = 3x + 5. Find f(4). Replace x with 4: f(4) = 3(4) + 5 = 12 + 5 = 17.

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f(3) = 2(3) − 7 = 6 − 7 = −1. f(0) = 2(0) − 7 = 0 − 7 = −7. f(0) gives the y-intercept of the function.

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Replace x with −2: h(−2) = (−2)² − 4(−2) + 1 = 4 + 8 + 1 = 13. Key: (−2)² = 4 (positive). −4(−2) = +8 (negative times negative = positive).

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The error is in −4². When substituting a negative number, use brackets: (−4)² = +16. Without brackets: −4² = −16 (squaring only 4, then negating — wrong). Correct: f(−4) = (−4)² + 3 = 16 + 3 = 19.

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The vertical line test: draw (or imagine) any vertical line through a graph. If every vertical line crosses the graph at most once → the graph represents a function. If any vertical line crosses the graph more than once → it is NOT a function (one x has two y-values).

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No — a vertical line through the centre of the circle crosses it twice (two y-values for one x). Since one input (x) gives two outputs (y), the circle fails the vertical line test and is not a function.

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Yes — every vertical line crosses the V-shape exactly once. Although the V looks like two lines meeting at a point, each x-value still gives exactly one y-value. y = |x| passes the vertical line test and is a function.

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No — this describes a one-to-one function (injective), not just any function. The vertical line test only checks if each x gives at most one y. It is fine for two different x-values to produce the same y (many-to-one is still a function).