FUNCTIONS (LIVE)

04 MAY 2015

Section A: Summary Notes Functions can be one-to-one relations or many-to-one relations. A many-to-one relation associates two or more values of the independent (input) variable with a single value of the dependent (output) variable. The domain is the set of values to which the rule is applied (A) and the range is the set of values (also called the images or function values) determined by the rule. Example of a one-to-one function: 𝑦 = 𝑥 + 1

Example of a many-to-one function: 𝑦

= 𝑥2

Function Notation For the function y = f (x), y is the dependent variable, because the value of y (output) depends on the value of x (input). We say x is the independent variable, since we can choose x to be any number. Similarly, if g (t) = 2t + 1, then t is the independent variable and g is the function name.

Vertical Line Test Given the graph of a relation, there is a simple test for whether or not the relation is a function. This test is called the vertical line test. If it is possible to draw any vertical line (a line of constant x) which crosses the graph of the relation more than once, then the relation is not a function. If more than one intersection point exists, then the intersections correspond to multiple values of y for a single value of x (one-to-many). If any vertical line cuts the graph only once, then the relation is a function (one-toone or many-to-one).

Inverse Functions An inverse function is a function which does the “reverse” of a given function, it is the reflection of the -1 graph in the line 𝑦 = 𝑥. More formally, if f is a function with domain x, then f is its inverse function if -1 and only if f (f (x)) = x for every x 2 X. y = f(x): indicates a function -1

f (y) = x : indicates the inverse function A function must be a one-to-one relation if its inverse is to be a function. If a function f has an inverse -1 function f , then f is said to be invertible -1

Given the function f(x), we determine the inverse f (x) by:   

interchanging x and y in the equation; making y the subject of the equation; expressing the new equation in function notation.

Note: if the inverse is not a function then it cannot be written in function notation. For example, the inverse of 𝑓 𝑥 = 3𝑥 2 cannot be written as 𝑓 −1 𝑥 = ± inverse as 𝑦 = ±

1 3

1 3

𝑥 as it is not a function. We write the

𝑥 and conclude that f is not invertible.

Logarithms A logarithm is a way of writing an exponential equation with the exponent as the subject of the formula. A logarithmic function is the inverse function of the exponential function. If 𝑥 = 𝑏 𝑦 𝑡𝑕𝑒𝑛 𝑦 = 𝑙𝑜𝑔𝑏 𝑥 𝑤𝑕𝑒𝑟𝑒 𝑏 ≠ 1 𝑎𝑛𝑑 𝑥 > 0

Section B: Exam practice questions Question 1 In the diagram, the graphs of the following functions have been sketched:

f ( x)  a ( x  p ) 2  q

and

g ( x) 

a q x p

The two graphs intersect at A(2 ; 4) and the turning point of the parabola lies at the point of intersection of the asymptotes of the hyperbola. The line parabola.

x 1

is the axis of symmetry of the

A(2 ; 4)

T

y2

x 1

f ( x)

1.1

Determine the equation of

in the form

1.2

Determine the equation of g ( x) in the form

1.3

Write down the range for the graph of f.

1.4

Write down the values of x for which

y  a( x  p)2  q

(3)

a q x p

(3)

y

g ( x)  0

(1) (2)

Question 2 Given:

f ( x)  2( x  1)2  8

2.1

Sketch the graphs of h and f

and

h( x )  4 x

Indicate ALL intercepts with the axes and any turning points. 2.2