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Maths revision course by Miriam Hanks
1
Functions and GraphsYou should be able to sketch these graphs:y = x2 and y = x3
y = x2
y = x3
Maths revision course by Miriam Hanks
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Functions and graphsYou should be able to sketch these graphs:y = and y = tan x
x
1
y = 1/x
y = tan x
Maths revision course by Miriam Hanks
3
Functions and graphsYou should be able to sketch these graphs:y = sin x and y = cos x
y = sin x y = cos x
Maths revision course by Miriam Hanks
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Functions and graphsYou should be able to sketch these graphs:y = ex and y = log x
y = ex
y = log x
Maths revision course by Miriam Hanks
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Functions – Domain and Range What are domain and range?
Domain is all possible x - values Range is all possible y-values
eg What are the domain and range of y = ex ?
The domain is x ε R, and the range is f(x) > 0.
Maths revision course by Miriam Hanks
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Functions – Domain and Range The domain of a fraction is restricted:
Since we can NEVER divide by zero, the denominator of a fraction cannot be zero, so for example, y = (3x + 2) / (x – 1) is restricted because
x cannot be equal to 1, ie the domain isx = 1
Maths revision course by Miriam Hanks
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Inverse functions f –1 (x)
eg Find the inverse function of f(x) = 2x + 1 To find an inverse
function:
swap x and y, then rearrange it to make y the subject
Swap x and y:
x = 2y + 1then rearrange to make y the subject:
y = (x – 1) /2
So f –1(x) = (x – 1 )/2
Maths revision course by Miriam Hanks
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Inverse functions f –1 (x)
eg A function f(x) = x2
has range f(x) > 0.
What is the domain of the inverse function ?
To find the domain and range of an inverse function:
swap the domain and range of the original function
x > 0
Maths revision course by Miriam Hanks
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Inverse functions f –1 (x)
To sketch an inverse function:
reflect the original function in the diagonal line y = x
f(x)
f -1(x)
y = x
Maths revision course by Miriam Hanks
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Composite functions f(x ) = x2 + 4x g(x) = 2x + 3
What is fg(x) ?
To find fg(x), pick up g and put it into f in place of each x.
fg(x) = (2x + 3)2 + 4(2x +3)
= 4x2 + 12x + 9 + 8x + 12
= 4x2 + 20x + 21
Maths revision course by Miriam Hanks
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Transformations of functions
f(x) + 5 moves UP 5 units
f(x + 5) moves LEFT 5 units
y = x2
y = x2 + 5
y = (x + 5)2
Maths revision course by Miriam Hanks
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Transformations of functions
- f(x)reflects in the x- axis
f(-x) reflects in the y-axisy = ex
y = e-x
y =-ex
Maths revision course by Miriam Hanks
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Transformations of functions
2f(x) multiplies the y-coordinates by 2
f(2x) divides the x-coordinates by 2 (and so you have twice as many waves)
y = sin xy = sin (2x)
y =2 sin (x)
