Maths revision course by Miriam Hanks

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Presentation transcript:

Maths revision course by Miriam Hanks Functions and Graphs You should be able to sketch these graphs: y = x2 and y = x3 y = x2 y = x3 Maths revision course by Miriam Hanks

Maths revision course by Miriam Hanks Functions and graphs You should be able to sketch these graphs: y = and y = tan x y = tan x y = 1/x Maths revision course by Miriam Hanks

Maths revision course by Miriam Hanks Functions and graphs You 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

Maths revision course by Miriam Hanks Functions and graphs You should be able to sketch these graphs: y = ex and y = log x y = log x y = ex Maths revision course by Miriam Hanks

Maths revision course by Miriam Hanks 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

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 is x = 1 Maths revision course by Miriam Hanks

Maths revision course by Miriam Hanks 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 + 1 then rearrange to make y the subject: y = (x – 1) /2 So f –1(x) = (x – 1 )/2 Maths revision course by Miriam Hanks

Maths revision course by Miriam Hanks 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

Maths revision course by Miriam Hanks Inverse functions f –1 (x) y = x To sketch an inverse function: reflect the original function in the diagonal line y = x f(x) f -1(x) Maths revision course by Miriam Hanks

Maths revision course by Miriam Hanks 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

Maths revision course by Miriam Hanks Transformations of functions f(x) + 5 moves UP 5 units f(x + 5) moves LEFT 5 units y = x2 + 5 y = x2 y = (x + 5)2 Maths revision course by Miriam Hanks

Maths revision course by Miriam Hanks Transformations of functions - f(x)reflects in the x- axis f(-x) reflects in the y-axis y = e-x y = ex y =-ex Maths revision course by Miriam Hanks

Maths revision course by Miriam Hanks 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 (2x) y = sin x y =2 sin (x) Maths revision course by Miriam Hanks