Inverses Revision Notes for VCE SSCE Mathematical Methods

Inverses

Introduction to inverse functions

You already know that exponential functions and logarithmic functions are inverses of each other. Specifically, if $f(x) = a^x$ and $g(x) = \log_a x$ (where $a > 0$ and $a \neq 1$ ), then these functions "undo" each other. This relationship is incredibly useful when finding inverses of related functions and transforming equations.

infoNote

The inverse relationship between exponential and logarithmic functions means that each function reverses the operation of the other. This is why we can use logarithms to solve exponential equations and vice versa.

Key properties of inverses

Because exponential and logarithmic functions are inverses, we get two fundamental properties:

\log_a(a^x) = x \text{ for all } x \in \mathbb{R}

This means if you raise $a$ to the power of $x$ , then take the logarithm base $a$ of the result, you get back to $x$ .

a^{\log_a x} = x \text{ for all } x \in \mathbb{R}^+

This means if you take the logarithm base $a$ of $x$ , then raise $a$ to that power, you get back to $x$ (as long as $x$ is positive).

chatImportant

These properties work because the functions are inverses - they reverse each other's operations. Think of it as "log undoes exponential, exponential undoes log."

Finding the inverse of an exponential function

To find the inverse of a function, we use the fact that reflecting in the line $y = x$ swaps the coordinates. This means we swap x and y, then solve for y.

lightbulbExample

Worked Example: Finding the Inverse of an Exponential Function

Find the inverse of the function $f: \mathbb{R} \to \mathbb{R}$ , $f(x) = e^x + 2$ and state the domain and range of the inverse function.

Solution:

To find the inverse, we swap $x$ and $y$ in the equation $y = e^x + 2$ :

x = e^y + 2

Now solve for $y$ :

x - 2 = e^y

Taking the natural logarithm of both sides:

y = \log_e(x - 2)

Therefore, the inverse function has rule f^{-1}(x) = \log_e(x - 2).

Finding domain and range:

The domain and range swap between a function and its inverse:

Domain of $f^{-1}$ = range of $f = (2, \infty)$
Range of $f^{-1}$ = domain of $f = \mathbb{R}$

Note that the range of $f(x) = e^x + 2$ is $(2, \infty)$ because $e^x > 0$ for all $x$ , so $e^x + 2 > 2$ . This becomes the domain of the inverse function, which makes sense because you can only take the logarithm of positive numbers, and here we need $x - 2 > 0$ .

Rearranging equations with x as the subject

Sometimes we need to rearrange an equation to make a different variable the subject. This involves the same principles as finding inverses.

lightbulbExample

Worked Example: Rearranging with x as Subject

Rewrite the equation $y = 2\log_e(x) + 3$ with $x$ as the subject.

Solution:

Start with the equation:

y = 2\log_e(x) + 3

Subtract 3 from both sides:

y - 3 = 2\log_e(x)

Divide both sides by 2:

\frac{y - 3}{2} = \log_e x

Since logarithm and exponential are inverses, we can write $x$ as an exponential:

x = e^{\frac{y-3}{2}}

Finding the inverse of a logarithmic function

The process for finding inverses of logarithmic functions is similar - swap variables and solve.

lightbulbExample

Worked Example: Finding the Inverse of a Logarithmic Function

Find the inverse of the function $f: (1, \infty) \to \mathbb{R}$ , $f(x) = 2\log_e(x - 1) + 3$ . State the domain and range of the inverse.

Solution:

To find the inverse, solve $x = 2\log_e(y - 1) + 3$ for $y$ :

Subtract 3 from both sides:

x - 3 = 2\log_e(y - 1)

Divide both sides by 2:

\frac{x - 3}{2} = \log_e(y - 1)

Since logarithm and exponential are inverses:

y - 1 = e^{\frac{x-3}{2}}

Add 1 to both sides:

y = e^{\frac{x-3}{2}} + 1

Hence f^{-1}(x) = e^{\frac{x-3}{2}} + 1.

Finding domain and range:

Domain of $f^{-1}$ = range of $f = \mathbb{R}$
Range of $f^{-1}$ = domain of $f = (1, \infty)$

Rearranging complex exponential equations

We can use logarithms to solve for variables that appear in exponents.

lightbulbExample

Worked Example: Solving for a Variable in an Exponent

Rewrite the equation $P = Ae^{kt}$ with $t$ as the subject.

Solution:

P = Ae^{kt}

Take natural logarithms of both sides:

\log_e P = \log_e(Ae^{kt})

Use the logarithm laws to expand the right side:

\log_e P = \log_e A + \log_e(e^{kt})

Since $\log_e(e^{kt}) = kt$ :

\log_e P = \log_e A + kt

Subtract $\log_e A$ from both sides:

\log_e P - \log_e A = kt

Divide both sides by $k$ :

t = \frac{1}{k}(\log_e P - \log_e A)

Using the logarithm law $\log_a x - \log_a y = \log_a\left(\frac{x}{y}\right)$ :

t = \frac{1}{k}\log_e\left(\frac{P}{A}\right)

Important relationships for inverse functions

chatImportant

Key Relationships Between a Function and Its Inverse

For a function $f$ and its inverse $f^{-1}$ :

Domain of f^{-1} = range of f
Range of f^{-1} = domain of f

This makes sense because the inverse function "swaps" the inputs and outputs.

For exponential and logarithmic functions where $a > 0$ and $a \neq 1$ :

$f(x) = a^x$ and $g(x) = \log_a x$ are inverse functions
This means $g = f^{-1}$ and $f = g^{-1}$

bookmarkSummary

Key Points to Remember:

The functions $f(x) = a^x$ and $g(x) = \log_a x$ are inverses of each other (where $a > 0, a \neq 1$ )
Two key properties: $\log_a(a^x) = x$ for all $x$ , and $a^{\log_a x} = x$ for all positive $x$
To find an inverse function: swap x and y, then solve for y
Domain and range swap between a function and its inverse: domain of $f^{-1}$ equals range of $f$ , and vice versa
Use logarithms to solve for variables that appear in exponents

Inverses (VCE SSCE Mathematical Methods): Revision Notes

Inverses

Introduction to inverse functions

Key properties of inverses

Finding the inverse of an exponential function

Rearranging equations with x as the subject

Finding the inverse of a logarithmic function

Rearranging complex exponential equations

Important relationships for inverse functions

Explore VCE SSCE Mathematical Methods Model Answers by Topics

The Index Laws

Rational Indices

Graphs of Exponential Functions

Solving Exponential Equations and Inequalities

Using Logarithms to Solve Exponential Equations and Inequalities

Exponential Models and Applications

Logarithms

Graphing Logarithmic Functions

Determining Rules for Graphs of Exponential and Logarithmic Functions

Solution of Exponential Equations Using Logarithms

Inverses

Exponential Growth and Decay

Explore VCE SSCE Mathematical Methods Quizzes by Topics

The Index Laws

Rational Indices

Graphs of Exponential Functions

Solving Exponential Equations and Inequalities

Using Logarithms to Solve Exponential Equations and Inequalities

Exponential Models and Applications

Logarithms

Graphing Logarithmic Functions

Determining Rules for Graphs of Exponential and Logarithmic Functions

Solution of Exponential Equations Using Logarithms

Inverses

Exponential Growth and Decay

Explore VCE SSCE Mathematical Methods Flashcards by Topics

The Index Laws

Rational Indices

Graphs of Exponential Functions

Solving Exponential Equations and Inequalities

Using Logarithms to Solve Exponential Equations and Inequalities

Exponential Models and Applications

Logarithms

Graphing Logarithmic Functions

Determining Rules for Graphs of Exponential and Logarithmic Functions

Solution of Exponential Equations Using Logarithms

Inverses

Exponential Growth and Decay

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