Explain in words why a given function does not have an inverse
Functions and Inverses
What is an inverse?
An inverse function, written as f-1, “undoes” the action of the original function f. If f(x)=y, then f-1(y)=x. Think of it like a vending machine: you put in a number (the input), the machine gives you a product (the output). The inverse is the machine that tells you which number you must have put in to get that product back.
When does a function have an inverse?
A function has an inverse **iff** it is *one‑to‑one* (also called *injective*). That means no two different inputs give the same output.
How to check? The Horizontal Line Test
- Draw a horizontal line across the graph of f.
- If the line ever cuts the graph at more than one point, the function is **not** one‑to‑one.
- Only if every horizontal line meets the graph at most once does an inverse exist.
Why a Given Function Might Not Have an Inverse
- Multiple inputs map to the same output: e.g. f(x)=x^2 gives f(2)=4 and f(-2)=4.
- Graphically, the curve folds back on itself, so a horizontal line can intersect it twice.
- Without a unique output for each input, you cannot “reverse” the function.
Example 1: f(x)=x2
| x | f(x)=x2 |
|---|---|
| -3 | 9 |
| -2 | 4 |
| -1 | 1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
Notice how f(-2)=4 and f(2)=4. Because two different inputs give the same output, f(x)=x2 is not one‑to‑one and therefore has no inverse over all real numbers.
Example 2: f(x)=sin(x)
The sine function repeats every 2π radians. For example, sin(π/6)=1/2 and sin(5π/6)=1/2. Thus, a horizontal line at y=1/2 cuts the graph at two points, so sin(x) is not one‑to‑one on its entire domain.
How to Fix It: Restrict the Domain
If you limit the domain so that each output is produced only once, the function can get an inverse.
- For f(x)=x2, restrict to x≥0 or x≤0. Then the inverse is f-1(y)=√y or f-1(y)=−√y respectively.
- For sin(x), restrict to [−π/2, π/2]. The inverse is the arcsine function sin-1(y).
Exam Tips
🔍 Check the domain: If the function is defined for all real numbers, you still need to test for one‑to‑one.
📐 Use the horizontal line test: A quick visual check can save time.
📝 Write a short justification: Explain that two different inputs give the same output, so no inverse exists.
💡 Remember domain restrictions: If the question asks for an inverse, they may have already restricted the domain. Verify that the restriction makes the function one‑to‑one.
?? Practice with graphs: Draw the graph of the function and try the horizontal line test yourself.
Quick Summary
A function f has an inverse f-1 iff it is one‑to‑one. If any horizontal line intersects the graph more than once, the function is not one‑to‑one and therefore does not have an inverse over its full domain. Restricting the domain can sometimes create a one‑to‑one function that does have an inverse.
Revision
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