Prove that the function h(x)=1-.2 x^3 of Example 1(c) is...

Prove that the function $h(x)=1-.2 x^{3}$ of Example $1(c)$ is one-to-one by showing that it satisfies the definition: $$\text { If } a \neq b, \text { then } h(a) \neq h(b)$$ [Hint: Use the rule of $h$ to show that when $h(a)=h(b),$ then $a=b .$ If this is the case, then it is impossible to have $h(a)=h(b) \text { when } a \neq b .]$

Prove that the function $h(x)=1-.2 x^{3}$ of Example $1(c)$ is one-to-one by showing that it satisfies the definition:
$$\text { If } a \neq b, \text { then } h(a) \neq h(b)$$
[Hint: Use the rule of $h$ to show that when $h(a)=h(b),$ then $a=b .$ If this is the case, then it is impossible to have $h(a)=h(b) \text { when } a \neq b .]$

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Key Concepts

One-to-One (Injective) Functions

A function is called one-to-one, or injective, if different inputs produce different outputs. In other words, if f(a) equals f(b), then it must follow that a equals b. This concept is fundamental in understanding whether a function has an inverse, because only injective functions have well-defined inverses.

Algebraic Manipulation in Function Proofs

To prove that a function is injective, one typically assumes that f(a) = f(b) and then uses algebraic techniques to simplify and rearrange the equality. The goal is to eliminate common terms and isolate the variables to show that the equality forces a = b. This process often involves subtracting identical terms from both sides and using properties of arithmetic operations.

Proof by Contrapositive

An alternative approach to proving that a function is one-to-one is through the contrapositive method. Instead of directly showing that a ? b implies f(a) ? f(b), one shows that if f(a) = f(b), then it must be that a = b. This method is logically equivalent to the original definition and is particularly useful when the algebraic manipulation is more straightforward in one direction.

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