Final answer:
A one-to-one function is a function in which each input has a unique output. To determine if the given function n(x) = 2x³ + 7 is one-to-one, we need to check if different inputs will produce different outputs.
Step-by-step explanation:
A one-to-one function is a function in which each input has a unique output. To determine if the given function n(x) = 2x³ + 7 is one-to-one, we need to check if different inputs will produce different outputs. We can do this by comparing any two inputs, x1 and x2, and checking if n(x1) is equal to n(x2). Let's do the calculation:
n(x1) = 2(x1)³ + 7
n(x2) = 2(x2)³ + 7
If n(x1) = n(x2), then 2(x1)³ + 7 = 2(x2)³ + 7
Simplifying, we get:
2(x1)³ = 2(x2)³
Dividing both sides by 2 and taking the cube root, we have:
x1 = x2
Since the inputs x1 and x2 are equal, it means that n(x1) and n(x2) will also be equal, making the function one-to-one.