Functions revision

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Functions- revision

• Inverse Functions
• Transformations
• Asymptotes
• Reciprocal and exponential functions

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• INVERSE FUNCTION reversing a function,
  undoing.
• f -1 notates an inverse function. (not 1/f)

Find the inverse of f(x)4x-2
4
-2
x
4x-2
(x2)/4
x
/4
2
So f -1 (x)(x2)/4
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A neat little trick

• As always in maths, there is a trick to this
• Write function as a rule in terms of y and x.
• Swap x and y
• Rearrange to get in terms of y.

f(x) 5x 7
y 5x 7
x 5y 7
x -7 5y y (x-7)/5 f-1(x) (x-7)/5
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Activity

• Find the inverse function of
• Feeling clever?

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Reflecting..
FACT inverses of any function are a reflection
in the line yx
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y f(x - a)
y - b f(x)
y - b f(x - a)
y k f(x)
stretches by a factor k along the y-axis
y f(x/k)
stretches by a factor k along the x-axis
y - f(x)
reflects in the x-axis
y f(-x)
reflects in the y-axis
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Trig Graphs translation

• How will ysin x, ysin (x) 1, ysin (x)
  3 work in degrees.
• How is the ysin x transformed to make the other
  two graphs?

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y sin x - 3
-For ysin x 1 there is a translation of 1 unit
up. ( y - 1 sin x ) -For ysin x - 3 there is
a translation of -3 unit up. ( y 3 sin x
) -What about ycos x 2 or ytanx 4?
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Graphs more translation

• Plot using a graphic calculator and then sketch
  ysin x, ysin( x 90) and ysin(x-45).
• How is the ysin x transformed to make these two
  graphs?

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For ysin( x 90) there is a translation of -90
units in the x direction. For ysin( x - 45).
there is a translation of 45 units in the x
direction.
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Graphs Cosine

• Plot using a graphic calculator and then sketch
  ycos x, y2 cos x and ycos 1/4x.
• How is the ycos x transformed to make these two
  graphs?

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For y2 cos x there is a stretch of 2 in the y
direction. For ycos 1/4x there is a stretch of 4
in the x direction.
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y tan x asymptotes
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y0

• Asymptotes
• Lines of limit
• Never reaches them

x0
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Asymptotes
--gt Reciprocal Function
f(x) 1/x

• Asymptotes
• Lines of limit
• Never reaches them

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Asymptotes
--gt Shift 3 units parallel to x-axis
f(x) 1/x
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Asymptotes
--gt Shift 3 units parallel to x-axis
--gt Shift -2 units parallel to y-axis
f(x) 1/x
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Vertical 4x - 2 0 4x 2 x 0.5 is an asymptote
Solve the denominator 0 to get vertical
asymptote
Horizontal needs a little trick
--gt Divide numerator and denominator by x
as x gets very very big
Then (gets near to 3/4)
y 3/4 is an asymptote
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Equation Vertical Asymptotes
Horizontal Asymptotes
TOP TIPS
By solving the denominator 0, . you can
identify vertical asymptotes x ... i.e. the
value of x that makes y infinity 1/0 ?
Identify translations parallel to the yaxis to
find horizontal asymptote y .
You can always plot them on your GDC and manually
spot where they are.
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y 4x
5
y 4x
4
3
2
1
0
-3
-2
-1
0
1
2
3
-1
-2
-3
-4
-5
The x-axis is an asymptote
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y ax
y ax
0
Functions of the form f(x) ax are known as
exponential functions