To prove that the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 5x - 8$ is one-to-one, follow these steps:

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Proof

A function is one-to-one (or injective) if different inputs produce different outputs, i.e., $f(a) = f(b)$ implies $a = b$.

1. Start with the assumption $f(a) = f(b)$: $f(a) = 5a - 8, \quad f(b) = 5b - 8$ So, if $f(a) = f(b)$, then: $5a - 8 = 5b - 8$

2. Simplify the equation: Subtract $-8$ from both sides: $5a = 5b$

3. Divide through by 5: $a = b$

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Conclusion

Since $f(a) = f(b)$ implies $a = b$, the function $f(x) = 5x - 8$ is one-to-one. This proves the injectivity of $f$.

Would you like any further clarification or examples? 😊

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Follow-up Questions

1. What does it mean for a function to be one-to-one in real-world contexts?
2. How can you use derivatives to check if a function is injective?
3. What are examples of functions that are not one-to-one?
4. Can a one-to-one function also be onto (surjective)?
5. How does the linearity of $f(x) = 5x - 8$ contribute to its injectivity?

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Tip

When checking injectivity, always assume $f(a) = f(b)$ and see if it leads to $a = b$.