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Welcome to GCSE Edexcel Maths revision.

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Topic M 19: Functions and graph transformations.

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This video covers Higher tier.

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f of open bracket x close bracket denotes the output of function f for input x; it does not mean f multiplied by x.

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If f of open bracket x close bracket equals 3 x minus 2, then f of open bracket 4 close bracket equals 10.

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The domain is the permitted input set and the range is the resulting output set.

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For f of open bracket x close bracket equals the fraction with numerator open bracket 1 close bracket and denominator open bracket x close bracket ,

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end fraction ,

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x equals 0 is excluded.

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A composite function applies one function then another. f composed with g evaluated at open bracket x close bracket means f open bracket g of open bracket x close bracket close bracket : the function nearest x acts first.

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Worked example: f of open bracket x close bracket equals 2 x plus 1 and g of open bracket x close bracket equals x squared . f composed with g evaluated at open bracket 3 close bracket equals f of open bracket 9 close bracket equals 19,

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but g composed with f evaluated at open bracket 3 close bracket equals g of open bracket 7 close bracket equals 49.

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Composition order matters.

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The inverse function, f inverse, reverses a function; it is not its reciprocal.

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To find an inverse, write y equals f of open bracket x close bracket , solve for x in terms of y, then exchange the letters.

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Worked example: y equals 3 x minus 2 gives x equals the fraction with numerator open bracket y plus 2 close bracket and denominator open bracket 3 close bracket ,

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end fraction ,

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so inverse f of open bracket x close bracket equals the fraction with numerator open bracket x plus 2 close bracket and denominator open bracket 3 close bracket ,

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end fraction .

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Applying f and then inverse f restores an allowed input.

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A function must be one-to-one on its chosen domain to have an inverse function. x squared on all real inputs is not one-to-one;

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restricting to x is greater than or equal to 0 permits the inverse function given by square root of x.

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y equals f of open bracket x close bracket plus a moves the graph up a units; y equals f of open bracket x close bracket minus a moves it down a.

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Every y-coordinate changes by the same amount.

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y equals f of open bracket x minus a close bracket moves the graph right a units, whereas y equals f of open bracket x plus a close bracket moves it left a.

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The sign inside the function often causes mistakes.

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A parabola translated two right and three up

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Worked example: y equals open bracket x minus 2 close bracket squared plus 3 is y equals x squared translated 2 right and 3 up.

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Its turning point is open bracket 2, 3 close bracket .

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y equals minus f of open bracket x close bracket reflects a graph in the x-axis: open bracket x,

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y close bracket becomes open bracket x,

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minus y close bracket . y equals f of open bracket minus x close bracket reflects it in the y-axis: open bracket x,

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y close bracket becomes open bracket minus x,

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y close bracket .

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For y equals square root of x,

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the graph of y equals minus square root of x lies below the x-axis with the same non-negative inputs. y equals square root of open bracket minus x close bracket instead lies to the left and requires x is less than or equal to 0.

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Track a distinctive point and any asymptotes through a transformation.

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For y equals the fraction with numerator open bracket 1 close bracket and denominator open bracket x minus 2 close bracket ,

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end fraction plus 3,

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asymptotes move from x equals 0,

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y equals 0 to x equals 2,

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y equals 3.

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Check a translated point by substitution.

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A graph's equation, domain and intercepts must agree with the described movement.

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That completes Functions and graph transformations.

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Revisit the notes and test yourself on the revision website.