Constructing inverse relations and functions
An inverse relation is formed by swapping the coordinates of every point. It is an inverse function only when the original function is one-to-one on its stated domain, so a branch restriction may be needed
\[P=(a,b)\quad\Rightarrow\quad P^{-1}=(b,a)\]
- Swap the coordinates To form an inverse relation, interchange the first and second coordinates of every point \(\displaystyle P=(a,b)\quad\Rightarrow\quad P^{-1}=(b,a)\)
- Exchange domain and range The domain of the inverse is the range of the original relation, and its range is the original domain \(\displaystyle D_{f^{-1}}=R_f,\quad R_{f^{-1}}=D_f\)
- Test for one-to-one behaviour An inverse is a function only when no two different inputs of the original function have the same output \(\displaystyle f(x_1)=f(x_2)\quad\Rightarrow\quad x_1=x_2\)
- Rearrange and select a branch Swap x and y, solve for y, and keep the branch or restriction that makes the inverse a function \(\displaystyle f(x)=2x+3\quad\Rightarrow\quad f^{-1}(x)=\frac{x-3}{2}\)
Worked example
For the one-to-one function shown, find its inverse rule, domain and range
\[f(x)=3x-4,\quad x\in[-2,4]\]
- \(\displaystyle y=3x-4\)Write the function using y as the output coordinate
- \(\displaystyle x=3y-4\)Swap x and y to describe the inverse relation
- \(\displaystyle x+4=3y\quad\Rightarrow\quad y=\frac{x+4}{3}\)Rearrange the swapped equation to make y the subject
- \(\displaystyle f(-2)=3(-2)-4=-10,\quad f(4)=3(4)-4=8\)Evaluate the endpoints to find the original range
- \(\displaystyle f^{-1}(x)=\frac{x+4}{3},\quad D_{f^{-1}}=[-10,8],\quad R_{f^{-1}}=[-2,4]\)The inverse domain is the original range, and the inverse range is the original domain
Common mistakes
Using a reciprocal instead of an inverse
✕\(\displaystyle f(x)=2x+3\quad\Rightarrow\quad f^{-1}(x)=\frac{1}{2x+3}\)✓\(\displaystyle f(x)=2x+3\quad\Rightarrow\quad f^{-1}(x)=\frac{x-3}{2}\)
An inverse undoes a function by swapping inputs and outputs; it is not found by taking a reciprocalLeaving the domain and range unchanged
✕\(\displaystyle D_f=[-2,4],\quad R_f=[-10,8]\quad\Rightarrow\quad D_{f^{-1}}=[-2,4],\quad R_{f^{-1}}=[-10,8]\)✓\(\displaystyle D_f=[-2,4],\quad R_f=[-10,8]\quad\Rightarrow\quad D_{f^{-1}}=[-10,8],\quad R_{f^{-1}}=[-2,4]\)
Swapping coordinates exchanges the original domain and range for the inverse relationIgnoring a branch restriction
✕\(\displaystyle f(x)=x^2-1,\quad D_f=\mathbb{R}\quad\Rightarrow\quad f^{-1}(x)=\sqrt{x+1}\)✓\(\displaystyle f(x)=x^2-1,\quad D_f=[0,\infty)\quad\Rightarrow\quad f^{-1}(x)=\sqrt{x+1}\)
The unrestricted quadratic is not one-to-one, so the square-root inverse needs a one-sided original domainDifficulty levels
introSwap coordinates in a finite relation and state the exchanged domain and range
standardFind an inverse rule for a one-to-one linear or restricted nonlinear function
challengeChoose a valid branch and determine whether a proposed inverse is a function
examConstruct and verify an inverse while tracking restrictions, domain and range
Questions about this topic
What is the difference between an inverse relation and an inverse function?+
Every relation can be reversed by swapping coordinates, but the reversed relation is a function only when each input has at most one output
Why must a function be one-to-one for its inverse to be a function?+
If two different original inputs share one output, swapping the coordinates gives one inverse input with two outputs
How is a branch selected for a quadratic inverse?+
Restrict the original quadratic to a one-sided domain, then choose the square-root sign that matches that branch
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 9: Logarithmic calculus
In this chapter
Differentiate natural-logarithmic functions9A Logarithmic differentiation: Evaluate and determine function rules9A Logarithmic differentiation: Find the maximal domain of an algebraic function9A Logarithmic differentiation: Simplify and combine algebraic fractions9A Logarithmic differentiation: Expand and factor polynomial expressions9A Logarithmic differentiation: Evaluate a polynomial or recover a coefficient parameterApply logarithmic derivatives9B Applications of logarithmic derivatives: Form and analyse an inverse relation or functionFind the equation of a line from its graph or from its intercept and slopeForm an equation of a parallel or perpendicular line9B Applications of logarithmic derivatives: Solve a linear equation9B Applications of logarithmic derivatives: Evaluate a polynomial or recover a coefficient parameter9B Applications of logarithmic derivatives: Solve a higher-degree polynomial equation or inequality9B Applications of logarithmic derivatives: Find the maximal domain of an algebraic function9B Applications of logarithmic derivatives: Evaluate and determine function rulesAnti-differentiate reciprocal and linear-reciprocal functions9C Reciprocal and linear-reciprocal antiderivatives: Simplify and combine algebraic fractions9C Reciprocal and linear-reciprocal antiderivatives: Expand and factor polynomial expressions9C Reciprocal and linear-reciprocal antiderivatives: Solve a linear equationIntegration by recognition with logarithmic components9D Recognising logarithmic derivatives: Expand and factor polynomial expressions9D Recognising logarithmic derivatives: Round to decimal places and significant figuresFind areas involving reciprocal curvesFind the area under a logarithmic curve using its inverseLogarithmic models in context9E Areas, inverse functions and logarithmic models: Form and analyse an inverse relation or function9E Areas, inverse functions and logarithmic models: Solve a linear equation9E Areas, inverse functions and logarithmic models: Solve a higher-degree polynomial equation or inequalityPerimeter and area of standard shapes9E Areas, inverse functions and logarithmic models: Round to decimal places and significant figures