Functions, Core 3, Maths

X - Independant variable - Domain       Y - Dependant variable - Range

One-One Relationship - For each value of x there is a unique value of y. 

y is a function of x

Many-One Relationship - Many values of x map to one value of y.

y is a function of x

One-Many Relationship - One value of x maps to many values of y.

y is not a function of x, but x is a function of y

Many-Many Relationship - Many values of x map to many values of y.

y is not a function of x

gf(x)

g - second function applied

f - first function applied.

The composite function gf, can only be formed if the range of g is a subset of  (contained within) the domain of f.

Inverse functions are the reverse of the the original function.

f(x) = 2x+1

1----->3

2----->5

3----->7

4----->9

f ^-1 (x) = x-1/2

3----->1

5----->2

etc.

Even function - f(x) = f(-x) for all valuse of x - The graph is symmetrical about the y-axis.

Eg. x², cos x

Odd function - f(x) = -f(-x) for all values of x

The graph has 180 degrees rotational symetry about the origin. (0,0)

Eg. x, sin x

Most functions are neither even nor odd.

Eg. f(x) = x² + 1/x

The notation IxI means the magnitude of x, ignoring the sign.

I6I = 6  and I-7I = 7.

IxI = x if x > 0

IxI = -x if x < 0

To sketch y = If(x)I

• Sketch the curve y = f(x), using a dashed line for points below the x axis.
• Reflect any part of the curve below the x axis in the x axis.

To sketch y = f(IxI)

• Sketch y = f (x) for x greater or equal to 0.
• Reflect the parts of the curve to the right of the y axis in the y axis.

All these transformations apply to y = f(x) a>0

y=f(x) +a Translation (0,a)

Adding a to the function moves the graph a units up.

y = f(x) -a Translation (0,-a)

Subtracting a from the function the function moves the graph a units down.

y = f(x-a) Translation (a,0)

Subtracting a from x moves the graph a units to the right.

y=f(x+a) Translation (-a,0)

Adding a to x moves the graph a units to the left.

y=f(x-a)+b is a translation (a,b)