Core 3 - C3

Seite 1

Chapter 1 - Algebraic Fractions

Simplifying Fractions Using Cancelling

Fractions that have common factors on the numerator and denominator can be cancelled, by division.

When multiplying fractions, multiply the top/bottom of one fraction by the top/botton of the other.

When dividing fractions, multiply the reciprocal of the dividing fraction by the fraction being divided.

Remainder Theorem:

F(x) = Q(x) x divisor + remainder

The remainder will always be over the divisor.

Seite 2

Chapter 2 - Functions

What is a Function?

A one-to-one, or many-to-one mapping, from a given domain and range.

The domain is the number of possible values of X a function can take.

The range is the number of possible values of Y, or F(x), a function can take.

Example:

f(x) = 2x + 5, x > 2

Its range will then be:

f(2) = 4+5 = 9.

So, f(x) > 9.

Composite Functions

When a function takes another function as its input value of X.

So, fg(x):

Do g(x) first.

Put the value of g(x) into f(x).

Inverse Functions

To find an inverse, you set f(x) = y, then rearrange to get x in terms of y.

After, replace the value of x with f^-1(x), replace all values of y with x.

Example:

f(x) = 2x+5

y = 2x+5

x = (y-5)/2

f^-1(x) = (x-5)/2

The graph of an inverse function is a reflection in the line y = x.

Inverse functions only exist for one-to-one functions.

The domain of an inverse function is equal to the range of the normal function.

The range of an inverse function is equal to the domain of the normal function.

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Chapter 3 - Exponential and Log Functions

Exponential

Exponential graph - eˣ:

Passes through (0,1), as e^0 will always be 1.

Log Functions

Ln(x) is the inverse of eˣ.

Use log rules (from C2) to manipulate log expressions.

You can take the log values of both sides of an equation, to solve for x when you have eˣ.

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Chapter 4 - Numerical Methods

Finding Roots

In any interval where f(x) changes sign, this shows that the interval must contain a root of the equation where f(x) = 0.

Iteration can be used to find an approximation for this root, by rewriting f(x) to get x in terms of x, and repeatedly running values of x through the equation.

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Chapter 5 - Transforming Graphs of Functions

Modulus Function

The modulus of a value is its postive value.

You need to be able to sketch these graphs.

Rules:

If it is |f(x)|, then make all values of y positive.

If it is f(|x|), then reflect all values of y created from positive values of x in the y axis.

When solving equations with modulus, you should draw the graphs and look at the intersections:

If the point is on the original f(x), then keep the normal value without the modulus.

If the point is on the modulus of f(x), then use the negative of the normal value without the modulus.

Applying a Combination of Transformations

f(x + a) is a horizontal translation of -a.

f(x) + a is a vertical translation of +a.

f(ax) is a horizontal stretch of scale factor 1/a.

af(x) is a vertical stretch of scale factor a.

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Chapter 6 - Trigonometry

sec θ = 1/cos θ

cosec θ = 1/sin θ

cot θ = 1/tan θ