Let $f:\mathbb{R}\to \mathbb{R}$ be defined by $f(x)=\{\begin{array}{ll}{x}^2,& x\le 1\\ 2x+1,& x>1\end{array}$ Find the domain and the range of $f$.

Step-by-step solution

1. The definition provides a formula for every real $x$; therefore the domain is all real numbers, $\mathbb{R}$.
2. For $x\le 1$, the expression is $x^2$. Since $x$ can be any real number ≤1, $x^2$ takes all values from 0 up to arbitrarily large numbers (e.g., $x=-10$ gives 100). Hence this part contributes $[0,\infty )$ to the range.
3. For $x>1$, the expression is $2x+1$. As $x$ exceeds 1, $2x+1>3$; this part also produces values larger than 3, which are already contained in $[0,\infty )$.
4. Combining the two parts, the overall range is the union of the two sets, which is $[0,\infty )$.

Concept tested: Domain and range of a piecewise function

Published 2 September 2026 · How this solution was prepared and checked

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