Identifying One-to-One Functions - Free Printable

Great! Let's go through the problem step by step and explain why option d) $ f(x) = x^3 $ is the correct answer — a one-to-one function.

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🔍 What is a one-to-one function?

A function is one-to-one (injective) if each output (in the range) corresponds to exactly one input (in the domain).
In other words:
> If $ f(a) = f(b) $, then $ a = b $.
No two different inputs give the same output.

There’s a helpful visual test:
👉 The Horizontal Line Test:
If no horizontal line intersects the graph of the function more than once, then the function is one-to-one.

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✔ Let's examine each option:

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#### a) $ f(x) = x^4 + x^2 = x^2(x^2 + 1) $

- This is an even-degree polynomial with only even powers.
- It's symmetric about the y-axis.
- For example:
- $ f(1) = 1^4 + 1^2 = 1 + 1 = 2 $
- $ f(-1) = (-1)^4 + (-1)^2 = 1 + 1 = 2 $
- So $ f(1) = f(-1) $, but $ 1 \neq -1 $
- ✘ Not one-to-one

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#### b) $ f(x) = x^2 $

- Classic parabola.
- Again, symmetric about y-axis.
- $ f(2) = 4 $, $ f(-2) = 4 $
- So two different inputs → same output
- ✘ Not one-to-one

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#### c) $ f(x) = \cos x $

- Cosine is periodic: repeats every $ 2\pi $
- Example: $ \cos(0) = 1 $, $ \cos(2\pi) = 1 $
- Many inputs give the same output
- ✘ Not one-to-one

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#### d) $ f(x) = x^3 $

Let’s check this carefully.

- Graph: A smooth curve passing through the origin, increasing throughout.
- Try values:
- $ f(1) = 1 $
- $ f(-1) = -1 $
- $ f(2) = 8 $
- $ f(-2) = -8 $
- No two different inputs give the same output.
- Also, it passes the horizontal line test: any horizontal line crosses the graph at most once.
- You can also prove algebraically:
Suppose $ f(a) = f(b) $
Then $ a^3 = b^3 $
Taking cube roots: $ a = b $
✔ So it is one-to-one

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📈 Visual Support from Diagrams:

- The graph of $ y = x^3 $ shown in blue is strictly increasing, so no horizontal line hits it more than once.
- The mapping diagram shows that each input maps to a unique output, and vice versa.
- The purple lines (horizontal) cross the curve only once → satisfies the horizontal line test.

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✔ Final Answer:

$$
\boxed{\text{d) } f(x) = x^3}
$$

This is the only one-to-one function among the choices.

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💡 Key Takeaway:

- Even functions (like $ x^2 $, $ x^4 $) are not one-to-one over all real numbers due to symmetry.
- Odd functions like $ x^3 $, especially strictly increasing ones, often are one-to-one.
- Always apply the horizontal line test or check if $ f(a) = f(b) \Rightarrow a = b $.