Unit 3 · Topic 2.8
Inverse Functions
Determine whether an inverse exists and construct it by reversing inputs and outputs on a restricted domain when needed. Develop the idea through symbolic, numerical, graphical, and contextual representations.
Learning Goals
- Interpret inverses as reversed input-output pairs.
- Determine one-to-one domains.
- Construct and verify inverse functions.
1. Essential Structure
Determine whether an inverse exists and construct it by reversing inputs and outputs on a restricted domain when needed.
Read the formula together with its domain, units, starting input, and the interval length over which change is measured.
2. Core Ideas
- If \(f(a)=b\), then \(f^{-1}(b)=a\).
- Domain and range exchange roles.
- A one-to-one function passes the horizontal-line test.
- Inverse graphs reflect across \(y=x\); \(f^{-1}\) is not \(1/f\).
3. Graph and Representation
Use graphical evidence together with an algebraic or numerical reason. A graphing window alone does not establish domain, end behavior, or model validity.
4. Original Worked Example
Find the inverse of \(f(x)=3x-5\).
Check the result in the original representation and explain its meaning, including units when the quantities are contextual.
5. AP Reasoning Workflow
- Check one-to-one behavior or restrict the domain.
- Write \(y=f(x)\).
- Swap \(x\) and \(y\).
- Solve for \(y\) and state the new domain.
- Verify by composition.
A strong AP response shows the mathematical evidence first and then states a precise conclusion.
6. Extended Example and Application
For \(f(x)=(x-2)^2+1\) restricted to \(x\ge2\), the inverse is \(f^{-1}(x)=2+\sqrt{x-1}\), with domain \(x\ge1\).
7. Technology and Validation
Graph the function, inverse, and \(y=x\) with equal axis scales as a visual verification.
When technology is used, record the model or result, preserve sufficient precision, and explain why the output answers the question.
8. Understand an Inverse as a Reverse Mapping
An inverse function reverses each input-output assignment of the original function. If
then, on an invertible domain,
The notation \(f^{-1}\) names a new function; it does not mean the reciprocal \(1/f\).
| Original statement | Reversed statement |
|---|---|
| Input 4 maps to output 11 | Input 11 maps back to output 4 |
| \((4,11)\) lies on \(f\) | \((11,4)\) lies on \(f^{-1}\) |
| Domain of \(f\) | Range of \(f^{-1}\) |
| Range of \(f\) | Domain of \(f^{-1}\) |
Reversing a process is possible as a function only when every allowed output identifies exactly one original input.
9. Decide Whether a Function Is Invertible
A function is one-to-one on a stated domain when distinct inputs produce distinct outputs. Graphically, it must pass the horizontal line test: no horizontal line may intersect the graph more than once.
| Function | Natural-domain behavior | Invertible? |
|---|---|---|
| \(2x-7\) | Strictly increasing | Yes, on all real numbers |
| \(x^3\) | Strictly increasing | Yes, on all real numbers |
| \(x^2\) | Decreases, then increases | No, unless the domain is restricted |
| \(|x|\) | Two inputs often share one output | No, unless one branch is selected |
| Constant function | Every input shares one output | No on any interval containing multiple points |
A convenient restriction should keep one complete branch and match the context. For \(x^2\), either \(x\ge0\) or \(x\le0\) is valid, but the two choices produce different inverse formulas.
10. Find and Evaluate an Inverse from a Table
Construct an inverse table by exchanging every input-output pair.
| \(x\) | \(f(x)\) | Inverse pair |
|---|---|---|
| -2 | 7 | \(f^{-1}(7)=-2\) |
| 0 | 10 | \(f^{-1}(10)=0\) |
| 3 | 16 | \(f^{-1}(16)=3\) |
| 5 | 21 | \(f^{-1}(21)=5\) |
The original outputs \(7,10,16,21\) become inverse inputs. If an output appeared twice in the original table with two different inputs, reversing the pairs would assign two outputs to one inverse input; that would show the original function is not one-to-one on the listed domain.
11. Reflect the Graph across the Identity Line
Exchanging \(x\) and \(y\) reflects every point across \(y=x\). Therefore, if \((a,b)\) lies on \(f\), then \((b,a)\) lies on \(f^{-1}\).
| Feature of \(f\) | Corresponding feature of \(f^{-1}\) |
|---|---|
| x-intercept \((a,0)\) | y-intercept \((0,a)\) |
| y-intercept \((0,b)\) | x-intercept \((b,0)\) |
| Horizontal asymptote \(y=k\) | Vertical asymptote \(x=k\) |
| Vertical asymptote \(x=h\) | Horizontal asymptote \(y=h\) |
| Domain interval | Range interval |
Use equal scales on both axes when checking reflection visually. A stretched coordinate window can make correct inverse graphs look nonsymmetric.
To estimate \(f^{-1}(6)\) from the original graph, find the point whose output is 6 and read its input; do not simply evaluate \(f(6)\).
12. Construct an Inverse Algebraically
For a one-to-one function, write \(y=f(x)\), exchange the roles of input and output, and solve for the new output. Consider
- Write \(y=(3x-5)/(2x+1)\).
- Swap the variables: \(x=(3y-5)/(2y+1)\).
- Clear the denominator: \(2xy+x=3y-5\).
- Collect the y-terms: \(y(2x-3)=-x-5\).
- Solve: \(y=(x+5)/(3-2x)\).
The excluded value \(3/2\) is the horizontal asymptote and missing range value of \(f\); it becomes the excluded input of \(f^{-1}\). State restrictions even when the final formula appears familiar.
13. Restrict a Quadratic and Choose the Correct Branch
The function \(f(x)=(x-3)^2+2\) is not one-to-one on all real numbers. Restricting its domain selects one side of the vertex.
| Restricted domain of \(f\) | Inverse | Domain of inverse | Range of inverse |
|---|---|---|---|
| \([3,\infty)\) | \(f^{-1}(x)=3+\sqrt{x-2}\) | \([2,\infty)\) | \([3,\infty)\) |
| \(( -\infty,3]\) | \(f^{-1}(x)=3-\sqrt{x-2}\) | \([2,\infty)\) | \(( -\infty,3]\) |
The symbol \(\pm\) cannot remain in an inverse-function formula because it would assign two outputs to most inputs. The original domain restriction determines which branch is valid.
For instance, on \(x\ge3\), \(f(7)=18\) and \(f^{-1}(18)=3+4=7\). On the other branch, the same original output 18 reverses to \(-1\).
14. Verify and Interpret an Inverse
Two functions are inverses on their stated domains when both compositions return the identity:
For the rational function in the previous section, substituting \(f^{-1}(x)=(x+5)/(3-2x)\) into \(f\) simplifies to \(x\) for \(x\ne3/2\); reversing the composition gives \(x\) for \(x\ne-1/2\). The two restrictions differ because each composition starts in a different domain.
Context example: During a 15-minute test, water depth is modeled by
Its inverse is
The original model maps elapsed minutes to centimeters; the inverse maps a measured depth to elapsed minutes. For example, \(H^{-1}(18.4)=10\), so the water reaches 18.4 cm after 10 minutes.
AP-ready conclusion: report the inverse formula, exchange the domain and range, identify the reversed units, and retain any mathematical or contextual restrictions.
15. Common Errors
- Do not write \(f^{-1}(x)\) as \(\frac{1}{f(x)}\).
- Inverting a quadratic without restricting its domain.
- Keeping both square-root branches and producing a nonfunction relation.
- Giving a numerical result without a domain check, units, or interpretation.
Key Takeaways
- Determine whether an inverse exists and construct it by reversing inputs and outputs on a restricted domain when needed.
- Core relationship: \(f(f^{-1}(x))=x\)
- Error check: Do not write \(f^{-1}(x)\) as \(\frac{1}{f(x)}\).
- Find and verify the inverse of \((5x+2)/3\).
- Give two restrictions of \(x^2-4\) and their corresponding inverses.
- Explain how inverse domain and range are determined.