C3 – Functions Summary
C3 – Functions Summary
 Function f(x): one-to-one or many-to-one mapping (for each value of x
there is only one value of y)
 Domain: set of numbers for which the function is defined (all possible x
values)
 Range: set of values the function can take for a given domain (all
possible y values)
 Composite function: a function of a function. gf(x) = g(f(x)) – f first then
g. Domain of g = range of f.
 Inverse function f-1(x): exists for one-to-one mappings only (for every x
there is only one y, and for every y there is only one x). For a function to
have an inverse its domain may need to be restricted. Domain of f(x) =
range of f-1(x) and vice versa. ff-1(x) = f-1f(x) = x. On a graph reflect f(x) in
y = x to find f-1(x) – axes must have the same scale! To find an inverse
function swap x and y and rearrange to y =.
 Inverse trig functions: in radians to give same scale on axes, domains
restricted to create on-to-one mappings
 Function f(x): one-to-one or many-to-one mapping (for each value of x
there is only one value of y)
 Domain: set of numbers for which the function is defined (all possible x
values)
 Range: set of values the function can take for a given domain (all
possible y values)
 Composite function: a function of a function. gf(x) = g(f(x)) – f first then
g. Domain of g = range of f.
 Inverse function f-1(x): exists for one-to-one mappings only (for every x
there is only one y, and for every y there is only one x). For a function to
have an inverse its domain may need to be restricted. Domain of f(x) =
range of f-1(x) and vice versa. ff-1(x) = f-1f(x) = x. On a graph reflect f(x) in
y = x to find f-1(x) – axes must have the same scale! To find an inverse
function swap x and y and rearrange to y =.
 Inverse trig functions: in radians to give same scale on axes, domains
restricted to create on-to-one mappings
function
function
sin x
cos x
domain
-/2  x
 /2
0x
range
inverse
domain
-1f(x)1
sin-1x
-1f(x)1
(arcsin x)
-1f(x)1
cos-1x
-1f(x)1
range
-/2 
x  /2
0x
sin x
cos x
domain
-/2  x
 /2
0x
range
inverse
domain
-1f(x)1
sin-1x
-1f(x)1
range
-/2 
-1f(x)1
0x
-f(x)
-/2  x  /2
(arcsin x)
-1f(x)1
(arcos x)
-/2  x  /2
(arcos x)
tan-1x
-f(x) -/2  x  /2
(arctan x)
 Modulus function: positive value of the function, sometimes called
absolute value. On a graph reflect above the x axis. When solving
equations or inequalities refer to a graph, consider separate cases
where values within the modulus sign are positive and negative.
 Transformations of functions:
tan x
-f(x)
a
 
b
f(x – a) + b
translate f(x) by vector
-f(x)
f(-x)
af(x)
reflect f(x) in x axis
reflect f(x) in y axis
stretch f(x) by scale factor a parallel to y axis
(everything a times as far from x axis as before)
f(bx)
stretch f(x) by scale factor
(everything
cos-1x
x  /2
1
parallel to x axis
b
1
times as far from y axis as before)
b
tan x
-/2  x  /2
-f(x)
tan-1x
(arctan x)
 Modulus function: positive value of the function, sometimes called
absolute value. On a graph reflect above the x axis. When solving
equations or inequalities refer to a graph, consider separate cases
where values within the modulus sign are positive and negative.
 Transformations of functions:
a
 
b
f(x – a) + b
translate f(x) by vector
-f(x)
f(-x)
af(x)
reflect f(x) in x axis
reflect f(x) in y axis
stretch f(x) by scale factor a parallel to y axis
(everything a times as far from x axis as before)
f(bx)
stretch f(x) by scale factor
(everything
1
parallel to x axis
b
1
times as far from y axis as before)
b