How and when can functions be combined?

Key questions

  1. 1

    What are the domain and range of a function?

  2. 2

    What happens if we add, subtract, multiply or divide functions?

  3. 3

    What happens if we apply one function to the output of another?

  4. 4

    What do we mean by transformations of functions?

  5. 5

    When does a function have an inverse?

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Resource type Title
Building blocks Forming functions — defining domains
Scaffolded task Making inverse functions
Investigation Order! Order!


Resource type Title
Building blocks Composing gets me nowhere
Building blocks Invertibles
Fluency exercise Domain and range dominoes
Fluency exercise Function builder I
Package of problems Compose!
Package of problems Translating or not?
Many ways problem Transformers
Food for thought Name that graph again
Food for thought Picture the process II
Bigger picture Going backwards
Bigger picture It's a matter of perspective
Bigger picture Maths for dinner

Review questions

Title Ref
Can we compose these polynomial functions? R7553
Can we compose two simple functions? R9527
Can we draw the graph of $\left| x + [x] \right|$? R5362
Can we find an inverse and sketch a modulus function? R5646
Can we find the inverse of a function in three parts? R8028
Can we find the ranges of $f$ and $g$, and the function $f \circ g$? R7607
Can we identify the right sketch graph of this reciprocal function? R5868
Can we identify the right sketch of a product of functions? R8749
Can we identify the right sketch of a transformed function? R7119
Can we pick the graph of $y=\sin^2{\sqrt{x}}$? R6656
Can we sketch and describe these composite functions? R5595
Can we sketch the graph of $g \colon x \to x - [x]$? R5764
Can we sketch the inverse of this composite function? R9824
Can we sketch these four related curves? R8427
Can we solve this equation with a sum of square roots? R6653
Can we work with the domain, codomain and range of a bijection? R5966
For which $x$-values does a function equal its inverse? R6735
How do these transformations change the graph of $f(x)$? R9673
How many solutions does this composite function equation have? R9924
When does $f$ satisfy the identity $f(t)-f(1-t)=g(t)$? R9798
When does this trig function have its maximum value? R8411
Which of these function statements are true? R9523
Which transformations give us the graph of $y=2^{x^2-4x+3}$? R6607