Which of the following is not one-to-one on its natural domain? 𝑔(𝑥)=7𝑥+8 𝑝(𝑥)=7𝑥3+8 𝑓(𝑥)=7𝑥 ℎ(𝑥)=7𝑥2+8

Question

Which of the following is not one-to-one on its natural domain?

𝑔(𝑥)=7𝑥+8

𝑝(𝑥)=7𝑥3+8

𝑓(𝑥)=7𝑥

ℎ(𝑥)=7𝑥2+8

Answer 1

The first three functions are all increasing, so they all pass the horizontal line test and are therefore

one-to-one.

The fourth function is the equation of a parabola, which fails the horizontal line test and so is not

one-to-one.

Answer 2

If you are able to find x₁ and x₂ such that x₁ ≠ x₂ but f(x₁) = f(x₂), then the function is not one-to-one.

For the function h(x), Let x₁ = a and x₂ = -a, then h(x₁) = 7a² + 8 and h(x₂) = 7a² + 8.

Clearly x₁ ≠ x₂ but h(x₁) = h(x₂). Therefore, h(x) is not one-to-one