Functions

Objective Deadlines / Progress D

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Find the domain and range of a function;Draw graphs to find the domain and range

Identify odd and even functions;Sketch graphs of odd and even functions; use their properties to solve problems

Find the domain and range of reciprocal functions of the

form f ( x )=ax+bcx+d ;

and other functions such as y=e f ( X ) , y=ln f ( x ) ,y=sin f (x ), …

Find the domain and range of other functions such as y=e f ( x ) , y=ln f ( x ) , y=sin f (x ) , …

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Find inverse functions; know how to use a swapping method; find the domain and range of inverse functions;

Understand graphs of inverse functions as reflections of f(x) in the line y = x;

Find composite functions; know how to use a swapping method; find the domain and range of composite functions;

Mod

ulus

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tions

Understand the two transformations by modulus functions graphically; Sketch graphs of |f (x)| and f|x|

Solve equations (finding intersections of graphs) including modulus functions both by algebraic methods and by using graphs

Solve inequalities (i) algebraically by squaring both sides (ii) using graphical and algebraic methods together;

1

Functions

Notes functions

2

Functions

WB1 Domain and Range – graphicallystate the domain and range of each graph

WB2 Sketch the graph of y = f(x) and state the domain and range

a) f(x) = x2−4 x+3, −1<x≤ 4 b) f : x⟼( 2 x+1 ,−2<x<4¿13− x , 4 ≤ x<10)

3

Functions

WB3 Sketch each graph and state its domain and range

a) y=x2−6x+11

b) f ( x )=3 x+2x

c) g( x )= 1x2

d) h ( x )= 1(x−3)(x+2)

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Functions WB4 The function h(x) is defined by h ( x )=1

x+2, xϵ R x ≠0

a) Sketch a graph of h(x)b) Solve these equations h(x) = 3 h(x) = 4 h(x) = 1c) Explain why the equation h(x) = 2 has no solution

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Functions WB5 Rearrange each equation to find all the asymptotes. Sketch each graph and state its

domain and range

a¿ y=3 x+1x−1

b¿ y= 4 x+32x+1

6

Functions

WB6a Draw a sketch of the function defined by g(t) = 3t + 2, and state it’s domain and range

WB6b f(x) = √7−x , Sketchthe graph of y=f ( x)What is the domain of f ? What is the range of f ?

7

Functions

Odd and even functions

A mapping represents an even function if f ( x )=f (−x )

even function: Reflection of positive y values in the y-axis

Cos x is an even function

A mapping represents an odd function if f ( x )=−f (−x )

odd function: Rotation of positive y value 180 about centre (0, 0)

Sin x and tan x are odd functions

8

Functions

WB7 Draw a sketch of the function defined by f ( x )= 6x+1

, x1

State it’s domain and rangeIs f(x) an odd function? Give a reason for your answer

WB8 Determine whether the following functions are odd, even or neither:Use a graph package to check / investigate

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