# Functions. Mappings transform one set of numbers into another set of numbers. We could display a mapping in three ways

Functions

Mappings transform one set of numbers into another set of numbers. We could display a mapping in three ways

• y = 2x + 1

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y = x2                 is an example of a many-to-one mapping

y = ±√x        is an example of a one-to-many mapping

Functions only have one output for each input. They are one-to-one or many-to-one mappings.

Function Notation

f(x) = 2x + 1                xЄ R

It is called the domain.

f:x --> 2x + 1

The set of outputs is called the range.

f(x) = x2        xЄ R

is f(x) ≥ 0        or         f ≥ 0

BUT NOT x ≥ 0

We can think of the domain as the values on the x axis and the range as the values on the y axis.

It is often easiest to find the range by sketching the graph.

Sometimes we need to limit a domain so that all the inputs give an output (ie can't square root a negative number and cannot divide by zero.

f(n) means find the value of f(x) when x = n

Check that roots of the equation lie within the domain, otherwise reject them

Composite Functions

f(g(x)) we write as fg(x)

Do g(x) first then f(x) !!!

Note that in general fg(x) ≠ gf(x)

Inverse Functions

If a function is one-to-one then an inverse function exists, called f-1(x)

For Simple Functions

eg f(x) = 4x + 1, xЄR

x

Alternatively, swap y and x throughout and rearrange to make y the subject.

Many-to-one functions can be made into one-to-one functions by restricting the domain. f-1(x) can then be found

Range of function = Domain of Inverse Function

Domain of Function = Range of Inverse Function

Sketching the graph can be helpful to determine the range and the domain.

y = f(x) is a reflection of y = f-1(x) in the line y = x

An asymptote is formed where there is a "forbidden" number. For the domain, a vertical asymptote is formed. For the range, a horizontal asymptote is formed.

To find the inverse of a quadratic function, swap x and y and complete the square.

Algebraic Fractions

Simplifying

To simplify

• Look for common factor in the numerator and denominator
• Sometimes we have to factorise first
• If we have fractions in the numerator or denominator, multiply to remove the fraction

Multiplying and Dividing

• Multiply the numerators and multiply the denominators.
• Cancel where necessary
• Where one fraction is divided by another, invert/flip the second fraction and multiply.