Functions Notes functions

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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 ,−3<x<4¿13− x , 4 ≤ x<10)

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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

WB5 Draw a sketch of the function defined by g(t) = 3t + 2, and state it’s domain and range

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Functions WB6 f(x) = √7−x , Sketchthe graph of y=f ( x)

What is the domain of f ? What is the range of f ?

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

Notes inverse functions

Inverse function graphicallySketch on the same axes f(x) and its inverseSketch on the same axes g(x) and its inverse

WB Find the Inverse function: f−1(x)

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Functions

f ( x )= x2

3−5 ,¿x∈ℜ ¿

f ( x )= 63 x−2

, x≠ 23

Find the Inverse function: f ( x )= 3 x+22 x−5

f ( x )=sin (2 x+1 )

3−5 ,¿ x∈ℜ ¿

WB

V

WB

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Functions

WB 17 A gas meter indicates the amount of gas in cubic feet used by a consumer. The number of therms of heat from x cubic feet of gas is given by the function f where f(x) = 1.034x, x > 0A particular gas company’s charge in £ for t therms is given by the function g where g(t) = 15 + 0.4t

(i) Find the cost of using 100 cubic feet(ii) Find a single rule for working out the cost given the number of cubic

feet of gas used

WB 18

The functions f and g are defined by: f : x↦ x2 ,¿¿¿

¿

Find i) fg(x) ii) fg(0) iii) gf(x) iv) gf(1) v) ff(x) vi) ff(-2)vii) gg(x) viii) gg(-7)

WB 19

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Functions

WB 20

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Functions

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Functions Review. Transformations of graphs

Learn these and practice them

f(x) = x3

sketch f(3x)sketch f(x+2)sketch 4 f(x)

f(x) = 2x2 + 3sketch f(3x)sketch f(x+2)sketch 4 f(x)

f(x) = 2x3 + 4xsketch f(2x)sketch f(x – 1)sketch 3f(x)

f(x) = Sin xsketch f(2x)sketch f(x + 90)sketch 3 f(x)

f(x) = Cos xsketch – f(x)sketch f(x – 30)sketch 2 f(x)

f(x) = Tan xsketch f(2x)sketch f(x + 180)sketch f (– x)

(i) Shifts

(ii) Stretches

(iii) Reflections

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+A

(x, y) (x + A, y)

+A

(x, y)

(x , y + A)

×

(x, y)

(x , y)

×A

(x, y)

(x, Ay)

(x, – y)

(x, y)

(x, y)(-x, y)

f ( x )+A is a shift in the y direction

f ( x−A ) is a shift in the x direction

f ( Ax ) is a stretch by scale

factor

1A in the x direction

Af (x ) is a stretch by scale factor A in the y direction

f (−x ) is a reflection of the graph in the y axis

− f ( x ) is a reflection of the graph in the x axis

Functions

WB21Solve the Inequality |x−12|>|3 x|

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Functions

WB22 Given that f ( x )=|x| and g( x )=x+3Sketch the graphs of the composite functions fg ( x )∧gf (x )Indicating clearly which is which

WB2312

Functions

Solve the inequality |3 x−1|<|6 x−1|

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Functions

WB24a) Solve the equation |9 x2−61|=60b) Hence, or otherwise, solve the inequality |9 x2−61|≥ 60

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Functions WB25 f ( x )=x4−4 x❑−240

a) Show that there is a root of f(x) = 0 in the interval [-4, -3]b) Find the coordinates of the turning point on graph of y = f(x)

Given that f ( x )=(x−4)(x3+a x2+bx+c ) find the values of a,b and cd) Sketch the graph of y = f(x)e) Hence sketch the graph of y = f(x)

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