Home » VCE » Maths Methods U3 & 4 Master Notes » Inverse Functions

# Inverse Functions

## 1.4 Inverse Functions

#### Inverse Functions

• A function maps the domain to the range. An inverse function simply does the opposite.
• However, recall that a function must have the type of relation which is a one-to-one relation. Therefore, if the inverse function (which, yes, is a function) exists, the inverse must have a one-to-one relation too.
• This implies that for an inverse function to exist (might be over a certain subset of the domain), the function must be one-to-one (over that particular subset if applicable).
• We denote the inverse of f as f^{-1}. In particular, if f(x)=y, then f^{-1}(y)=x.
• Therefore, we can deduce that:

## 1.4 Logarithmic Functions

#### The Logarithmic Function

• The logarithmic function with base is defined as follows:

a^{x} = b is equivalent to log_{a}{b}=x, where a \in R^{+}\setminus \left \{ 1 \right \} .

Note:a \in R^{+}\setminus \left \{ 1 \right \}‘ means that could be any positive number, excluding 1.

• The expression log_{a}{y}=x is read as: ‘the logarithm of y to the base a is equal to x’.

Example

## 1.3 Exponential Functions [Free]

#### The Exponential Function

• The exponential function is defined as follows:

y=a^x, where a \in R^+\backslash {1}.

Note:a \in R^+\backslash {1}’ means that could be any positive number, excluding 1.

Example

9=3^{2}=3 \times 3 is equivalent to log_{3}9=2.

8=16^{\frac{3}{4}}=\sqrt[4]{16^{3}}=\sqrt{\sqrt{16 \times 16 \times 16}}.

Graph & Properties

Read More »1.3 Exponential Functions [Free]

## 1.2 Power Functions

#### Introduction to Power Functions

• Power functions are functions with the form f(x)=x^{n}, where n is a rational number.
• Some examples of power functions are:

f(x)=x^{2},\ f(x)=x^{4},\ f(x)=x^{\frac{1}{4}},\ f(x)=x^{-5},\ f(x)=x^{\frac{1}{3}}

• The expression x^n is read as: ‘x to the power of n’.

Drawing Power Functions (for Positive Integers) and its Graph Properties