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GCSEMathsAQAHigherFlashcards2. AlgebraFunctions

Functions (AQA GCSE Maths)

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  • What is a function?

    A function is a combination of one or more mathematical operations that takes a set of numbers and changes them into another set of numbers.

  • What does the notation  straight f open parentheses x close parentheses equals...   indicate?

    The notation  straight f open parentheses x close parentheses equals...  indicates a function straight f, where x is the input and straight f open parentheses x close parentheses is the output.

  • What is the input of a function?

    The input of a function is the number being put into the function, often referred to as  x.

  • What is the output of a function?

    The output of a function is the number coming out of the function, often referred to as straight f open parentheses x close parentheses or y.

  • How would you find the value of straight f open parentheses 3 close parentheses for a function straight f open parentheses x close parentheses?

    To find the value of straight f open parentheses 3 close parentheses for a function straight f open parentheses x close parentheses, substitute x equals 3 into the equation of the function.

    For example, if straight f left parenthesis x right parenthesis equals 2 x plus 1, then straight f left parenthesis 3 right parenthesis equals 2 open parentheses 3 close parentheses plus 1 equals 6 plus 1 equals 7.

  • How would you find the value of x if you knew that straight f open parentheses x close parentheses equals 15?

    To find the value of x if you know that straight f open parentheses x close parentheses equals 15:

    • Set the equation for straight f open parentheses x close parentheses equal to 15

    • Then solve for x

    For example, if straight f open parentheses x close parentheses equals 2 x plus 1 and straight f open parentheses x close parentheses equals 15, then

    2 x plus 1 equals 15 space space space rightwards double arrow space space space 2 x equals 14 space space space rightwards double arrow space space space x equals 7

  • Define a composite function.

    A composite function is a function that 'combines' two functions.

    The output of one function is used as the input for the other function.

  • What does the notation fg open parentheses x close parentheses mean?

    fg open parentheses x close parentheses is the notation for a composite function made up of functions straight f and straight g.

    It means "straight f is applied to the output of straight g open parentheses x close parentheses".

    I.e., first put x into the function straight g, and next put the output straight g open parentheses x close parentheses into the function straight f.

    Note that the function on the right (straight g) acts first, and the function on the left (straight f) acts second.

  • What does the notation gf open parentheses x close parentheses mean?

    gf open parentheses x close parentheses is the notation for a composite function made up of functions straight g and straight f.

    It means "straight g is applied to the output of straight f open parentheses x close parentheses".

    I.e., first put x into the function straight f, and next put the output straight f open parentheses x close parentheses into the function straight g.

    Note that the function on the right (straight f) acts first, and the function on the left (straight g) acts second.

  • How would you find the value of fg open parentheses 2 close parentheses?

    To find the value of fg open parentheses 2 close parentheses:

    1. First substitute x equals 2 into the equation for straight g open parentheses x close parentheses.

    2. Then substitute the output straight g open parentheses 2 close parentheses into the equation for straight f open parentheses x close parentheses.

  • Give the definition of an inverse function.

    An inverse function is a function that does the exact opposite operations of the function it came from.

  • What is the notation for the inverse function of the function straight f open parentheses x close parentheses?

    The notation for the inverse function of the function straight f open parentheses x close parentheses is straight f to the power of negative 1 end exponent open parentheses x close parentheses.

  • What are the steps to find the inverse function straight f to the power of negative 1 end exponent open parentheses x close parentheses of straight f open parentheses x close parentheses?

    To find straight f to the power of negative 1 end exponent open parentheses x close parentheses:

    1. Write straight f open parentheses x close parentheses as y italic equals open parentheses formula space in italic space x close parentheses.

    2. Swap x and y to get x italic equals open parentheses formula space in italic space y close parentheses.

    3. Rearrange to make y the subject.

    4. Replace y with straight f to the power of negative 1 end exponent open parentheses x close parenthesesto rewrite as straight f to the power of negative 1 end exponent open parentheses x close parentheses equals....

  • If you knew that straight f open parentheses 3 close parentheses equals 10, how would you find straight f to the power of negative 1 end exponent open parentheses 10 close parentheses?

    An inverse function 'undoes' the operations of the original function.

    So if straight f turns 3 into 10, then straight f to the power of negative 1 end exponent turns 10 back into 3.

    I.e., if straight f open parentheses 3 close parentheses equals 10, then straight f to the power of negative 1 end exponent open parentheses 10 close parentheses equals 3.

  • What is ff to the power of negative 1 end exponent open parentheses x close parentheses equal to?

    ff to the power of negative 1 end exponent open parentheses x close parentheses is a composite function of straight f and its inverse straight f to the power of negative 1 end exponent.

    An inverse function 'undoes' the operations of the original function (and vice versa).

    So ff to the power of negative 1 end exponent open parentheses x close parentheses equals x.

    (Note that straight f to the power of negative 1 end exponent straight f open parentheses x close parentheses equals x as well.)

  • True or false?

    Completing the square can help find inverse functions.

    True.

    Completing the square can help rearrange a quadratic expression in order to find its inverse function.

Previous:Forming & Solving EquationsNext:Coordinate Geometry

    1. Topic 1

      Number Operations

    2. Topic 2

      Prime Factors, HCF & LCM

    3. Topic 3

      Powers, Roots & Standard Form

    4. Topic 4

      Fractions

    5. Topic 5

      Percentages

    6. Topic 6

      Simple & Compound Interest, Growth & Decay

    7. Topic 7

      Fractions, Decimals & Percentages

    8. Topic 8

      Rounding, Estimation & Bounds

    9. Topic 9

      Surds

    10. Topic 10

      Using a Calculator

    1. Topic 1

      Introduction to Algebra

    2. Topic 2

      Algebraic Roots & Indices

    3. Topic 3

      Expanding Brackets

    4. Topic 4

      Factorising

    5. Topic 5

      Completing the Square

    6. Topic 6

      Algebraic Fractions

    7. Topic 7

      Rearranging Formulas

    8. Topic 8

      Algebraic Proof

    9. Topic 9

      Linear Equations

    10. Topic 10

      Quadratic Equations

    11. Topic 11

      Simultaneous Equations

    12. Topic 12

      Iteration

    13. Topic 13

      Forming & Solving Equations

    14. Topic 14

      Functions

    15. Topic 15

      Coordinate Geometry

    16. Topic 16

      Linear Graphs

    17. Topic 17

      Graphs of Functions

    18. Topic 18

      Equation of a Circle

    19. Topic 19

      Estimating Gradients & Areas under Graphs

    20. Topic 20

      Real-Life Graphs

    21. Topic 21

      Solving Inequalities

    22. Topic 22

      Graphing Inequalities

    23. Topic 23

      Transformations of Graphs

    24. Topic 24

      Sequences

    1. Topic 1

      Ratios

    2. Topic 2

      Problem Solving with Ratios

    3. Topic 3

      Direct & Inverse Proportion

    4. Topic 4

      Standard & Compound Units

    5. Topic 5

      Exchange Rates & Best Buys

    1. Topic 1

      Symmetry & Shapes

    2. Topic 2

      Angles in Polygons & Parallel Lines

    3. Topic 3

      Bearings, Scale Drawing, Constructions & Loci

    4. Topic 4

      Circle Theorems

    5. Topic 5

      Area & Perimeter

    6. Topic 6

      Circles, Arcs & Sectors

    7. Topic 7

      Volume & Surface Area

    8. Topic 8

      Congruence, Similarity & Geometrical Proof

    9. Topic 9

      Area & Volume of Similar Shapes

    10. Topic 10

      Pythagoras & Trigonometry

    11. Topic 11

      Sine, Cosine Rule & Area of Triangles

    12. Topic 12

      3D Pythagoras & Trigonometry

    13. Topic 13

      Vectors

    14. Topic 14

      Transformations

    1. Topic 1

      Introduction to Probability

    2. Topic 2

      Simple Probability Diagrams

    3. Topic 3

      Tree Diagrams

    4. Topic 4

      Combined & Conditional Probability

    1. Topic 1

      Averages, Ranges & Data

    2. Topic 2

      Statistical Diagrams

    3. Topic 3

      Histograms

    4. Topic 4

      Cumulative Frequency & Box Plots

    5. Topic 5

      Scatter Graphs & Correlation