Final answer:
To prove that f(x) = 3x + 1 is a one-to-one function, we show if f(x1) = f(x2), then x1 must equal x2. This is confirmed by algebraic manipulation, which demonstrates that their corresponding x values are indeed equal.
Step-by-step explanation:
Demonstrating that the Function f(x) = 3x + 1 is One-to-One
To prove that the function f(x) = 3x + 1 is one-to-one, we need to show that for every x₁ and x₂ in the domain, if f(x₁) = f(x₂), then x₁ = x₂. Let's suppose f(x₁) = f(x₂) for some x₁ and x₂ in the real numbers.
f(x₁) = 3x₁ + 1
f(x₂) = 3x₂ + 1
Set these two expressions equal to each other because f(x₁) = f(x₂):
3x₁ + 1 = 3x₂ + 1
To find if x₁ equals x₂, subtract 1 from both sides:
3x₁ = 3x₂
Next, divide both sides by 3:
x₁ = x₂
Since x₁ and x₂ are equal whenever f(x₁) = f(x₂), this confirms that the function f(x) = 3x + 1 is indeed one-to-one.
Additionally, as the function represents a line with a non-zero slope (3), it continuously increases as x increases, which reinforces that it is one-to-one.