Chapter 1 Summary: 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 

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

• 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

• 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)

• 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 

• 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

• 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 p