Chapter 1 Summary: Functions

A summary of the main points covered in chapeter 1 on functions of the IB standard maths course

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Introducing functions

  • A relation is a set of ordered pairs 
  • The domain is the set of all the first numbers (x-values) of the ordered pairs 
  • The range is the set of the second number (y-values) in each pair
  • A function is a relation where every x-value is related to a unique y-value 
  • A relation is a function if any verticle line is drawn will not intersect the graph more the once - the vertical line test 
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The domain and range of a relation on a Cartesian

Interval notation

  • Use round brackets (,) if the value is not included in the graph or whene the graph is undefined at that point (a hole, an asymptote or a jump) 
  • Use square brackets [,] if the value is included in the graph 
  • Set notation: 

{x:x<6}

This means that the set of x values are such that x is less that 6

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Function notation

  • f(x) is read as 'f of x' and means 'the value of function f at x'. 
  • Remember! f(x)=a is not the same as f(a) 
    • eg. Find f(7) of the function f(x) = x - 2

This means find the value of the function when x is 7, it does not mean that the value of the function is 7. 

f(x) = 7 - 2

f(x) = 5

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Composite functions

  • The composition of function f with the function g is written as f(g(x)), which is read as 'f of g of x', or (fog)(x), which is read as 'f composed with g of x'. 
  • A composite function applies one function to the result of another and is defined by (fog)(x) =f(g(x)
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Inverse functions

  • The inverse of a function f(x) is  f−1(x). It reverse the action of a function 

Functions f(x) and g(x) are inverse of one another if:

  •  
    • (fog)(x) = x for all the x values in the domain of g and
    • (gof)(x) = x for all the x values in the domain of f
  • You can use the horizontal line test to identify inverse functions. If a horizontal line crosses a function more than once, there is no inverse 
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The graphs of the inverse functions

  • The graph of the inverse of a function is a reflection of that function in the line y=x 
  • To find the inverse function algebraically, replace f(x) with y and solve y 
  • The function I(x)=x is the identity function. It leaves x unchanged. So fof−1= I
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Transformations of functions

  • f(x)+k translates f(x) vertically a distance of k units upwards
  • f(x)-k translates f(x) vertically a distance of k units downwards
  • f(x+k) translates f(x) horizontally a distance of k units to the left, where k>0
  • f(x-k) translates f(x) horizontally a distance of k units to the right, where k>0
  • -f(x) reflects f(x) in the x axis
  •   f(-x) reflects f(x) in the y axis  
  • f(qx) stretches f(x) horizontally with a scale factor 1/q
  • pf(x) stretches f(x) vertically with scale factor
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