Determine whether the function $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x)=e^x-3$ is one‑to‑one on the whole real line. Justify your answer.

Step-by-step solution

1. Write the definition of a one‑to‑one (injective) function: $f$ is one‑to‑one if $f(x_1)=f(x_2)$ implies $x_1=x_2$.
2. Assume $f(x_1)=f(x_2)$. Then $e^{x_1}-3=e^{x_2}-3$.
3. Add 3 to both sides to obtain $e^{x_1}=e^{x_2}$.
4. Since the exponential function $e^x$ is strictly increasing on $\mathbb{R}$, $e^{x_1}=e^{x_2}$ implies $x_1=x_2$.
5. Therefore the implication in the definition holds, so $f$ is one‑to‑one.
6. State the final conclusion: $f$ is one‑to‑one on $\mathbb{R}$.

Concept tested: Injectivity of exponential function

Published 7 September 2026 · How this solution was prepared and checked