Rearranging Functions: Complete Guide, Examples, and Practice Tool
Rearranging functions means changing the subject of an equation so that one variable or expression is isolated. This skill is essential in algebra, inverse functions, graph transformations, physics formulae, finance models, engineering calculations, and exam problem solving.
On this page, you will learn how to rearrange linear, quadratic, rational, exponential, logarithmic, and composite functions using clear step-by-step methods.
Rearranging Functions Practice Tool
Select a function type and see the rearranged form, key steps, and restrictions.
What Does Rearranging Functions Mean?
Rearranging a function means rewriting it so that a chosen variable becomes the subject. For example, if a function is written as:
\[ y = 2x + 5 \]
and we want to make \(x\) the subject, we reverse the operations:
\[ y - 5 = 2x \]
\[ x = \frac{y - 5}{2} \]
Why Rearranging Functions Is Important
1. Inverse Functions
To find an inverse function, you usually replace \(f(x)\) with \(y\), swap \(x\) and \(y\), then rearrange to make \(y\) the subject.
2. Physics and Science
Many formulae require rearrangement, such as \(v = u + at\), \(F = ma\), and \(E = mc^2\).
3. Engineering
Engineers rearrange equations to solve for force, pressure, voltage, resistance, velocity, area, and volume.
4. Exams
Rearranging formulae appears in algebra, functions, coordinate geometry, trigonometry, calculus, and modelling questions.
The Core Rule: Do the Same Thing to Both Sides
When rearranging equations, equality must be preserved. Whatever operation you apply to one side, you must apply to the other side.
\[ A = B \Rightarrow A + c = B + c \]
\[ A = B \Rightarrow \frac{A}{c} = \frac{B}{c}, \quad c \ne 0 \]
Order of Operations When Rearranging
Rearranging usually reverses the normal order of operations. If a variable is multiplied, added, squared, or placed inside a function, you undo the outermost operation first.
| Original Operation | Inverse Operation | Example |
|---|---|---|
| Addition | Subtraction | \(x + 5 = y \Rightarrow x = y - 5\) |
| Subtraction | Addition | \(x - 7 = y \Rightarrow x = y + 7\) |
| Multiplication | Division | \(4x = y \Rightarrow x = \frac{y}{4}\) |
| Division | Multiplication | \(\frac{x}{3} = y \Rightarrow x = 3y\) |
| Square | Square Root | \(x^2 = y \Rightarrow x = \pm\sqrt{y}\) |
| Exponential | Logarithm | \(a^x = y \Rightarrow x = \log_a(y)\) |
Example 1: Rearranging a Linear Function
Given:
\[ y = mx + c \]
Make \(x\) the subject:
\[ y - c = mx \]
\[ x = \frac{y - c}{m}, \quad m \ne 0 \]
Example 2: Rearranging a Quadratic Function
Given:
\[ y = ax^2 + c \]
Subtract \(c\):
\[ y - c = ax^2 \]
Divide by \(a\):
\[ \frac{y-c}{a} = x^2 \]
Take the square root:
\[ x = \pm\sqrt{\frac{y-c}{a}} \]
Example 3: Rearranging a Rational Function
Given:
\[ y = \frac{a}{x+b} \]
Multiply both sides by \(x+b\):
\[ y(x+b) = a \]
Divide by \(y\):
\[ x+b = \frac{a}{y} \]
Subtract \(b\):
\[ x = \frac{a}{y} - b \]
Example 4: Rearranging an Exponential Function
Given:
\[ y = ab^x \]
Divide by \(a\):
\[ \frac{y}{a} = b^x \]
Use logarithms:
\[ x = \log_b\left(\frac{y}{a}\right) \]
Example 5: Rearranging a Logarithmic Function
Given:
\[ y = \log_a(x) \]
Convert to exponential form:
\[ x = a^y \]
Diagram: Rearranging as Reverse Operations
Rearranging Functions and Inverse Functions
Rearranging functions is directly connected to finding inverse functions. To find an inverse:
- Write \(f(x)\) as \(y\).
- Swap \(x\) and \(y\).
- Rearrange to make \(y\) the subject.
- Write the result as \(f^{-1}(x)\).
Example:
\[ f(x) = 3x - 4 \]
Write:
\[ y = 3x - 4 \]
Swap:
\[ x = 3y - 4 \]
Rearrange:
\[ x + 4 = 3y \]
\[ y = \frac{x+4}{3} \]
So:
\[ f^{-1}(x) = \frac{x+4}{3} \]
Common Mistakes When Rearranging Functions
| Mistake | Why It Happens | Correct Approach |
|---|---|---|
| Dividing only one term instead of the whole side | Forgetting brackets | Use \(x = \frac{y-c}{m}\), not \(x = y - \frac{c}{m}\) |
| Forgetting \(\pm\) | Taking square roots too quickly | Use \(x = \pm\sqrt{y}\) when solving \(x^2=y\) |
| Ignoring restrictions | Missing domain conditions | Check denominators, square roots, and logarithms |
| Wrong logarithm base | Confusing exponential and log rules | If \(b^x=y\), then \(x=\log_b(y)\) |
Domain and Restrictions Matter
Rearranging a function can change the way restrictions appear. Always check:
- Denominators must not equal zero.
- Even roots require non-negative expressions in real-number problems.
- Logarithm inputs must be positive.
- Inverse functions may require restricted domains.
Exam Skills: Where Rearranging Functions Appears
Rearranging functions is commonly tested in algebra and functions topics across school mathematics courses. It is especially useful for:
- Changing the subject of a formula
- Finding inverse functions
- Solving equations
- Working with exponential and logarithmic models
- Interpreting graphs
- Solving real-world modelling questions
Practice Questions
- Rearrange \(y = 5x - 9\) to make \(x\) the subject.
- Rearrange \(y = 4x^2 + 1\) to make \(x\) the subject.
- Rearrange \(y = \frac{7}{x-3}\) to make \(x\) the subject.
- Rearrange \(y = 2 \cdot 3^x\) to make \(x\) the subject.
- Find the inverse of \(f(x)=\frac{x-2}{5}\).
Answers
- \(x = \frac{y+9}{5}\)
- \(x = \pm\sqrt{\frac{y-1}{4}}\)
- \(x = \frac{7}{y}+3\)
- \(x = \log_3\left(\frac{y}{2}\right)\)
- \(f^{-1}(x)=5x+2\)
Frequently Asked Questions
What is rearranging functions?
Rearranging functions means rewriting a function or equation so that a chosen variable becomes the subject.
Why do we rearrange functions?
We rearrange functions to solve for unknown variables, find inverse functions, simplify formulae, and apply equations in real-world problems.
Is rearranging functions the same as solving equations?
They are closely related. Solving equations finds values, while rearranging changes the form of an equation to isolate a variable.
How do you rearrange a linear function?
Use inverse operations. For \(y=mx+c\), subtract \(c\), then divide by \(m\), giving \(x=\frac{y-c}{m}\).
Why do quadratic functions sometimes have two rearranged answers?
Because taking a square root can produce both positive and negative values, such as \(x=\pm\sqrt{y}\).
What is the connection between rearranging and inverse functions?
To find an inverse function, you swap \(x\) and \(y\), then rearrange to make \(y\) the subject.
What is the hardest part of rearranging functions?
The hardest parts are usually handling fractions, square roots, logarithms, and restrictions on the domain.
Can every function be rearranged?
Not always into a simple form. Some functions require advanced methods or cannot be rearranged using elementary algebra.
Conclusion
Rearranging functions is one of the most useful algebra skills. It helps students understand inverse functions, solve equations, manipulate formulae, and apply mathematics in science, engineering, economics, and technology. The best strategy is to identify the operations acting on the variable, undo them in reverse order, and always check restrictions such as division by zero, square root domains, and logarithm inputs.