Solving f(x)=3 piecewise

Problem

Solve $f(x)=3$ with $f(x)=\begin{cases} 5-x^2 & x\le 0 \\ x+2 & x>0\end{cases}$.

Solution

Solve the equation piece by piece, checking that the solutions fall in the interval of the respective piece.

Piece $x\le 0$: $5-x^2=3 \iff x^2=2 \iff x=\pm\sqrt{2}$. Only $x=-\sqrt{2}\approx -1{,}41$ satisfies $x\le 0$ ($+\sqrt{2}$ is discarded).

Piece $x>0$: $x+2=3 \iff x=1$, which satisfies $x>0$. ✓

Solutions: $\boxed{x=-\sqrt{2} \ \lor\ x=1}$.

Links

Topics: Functions and properties
Concepts: Piecewise function
Skills: Reasoning by cases · Solving equations
Exercise types: Solving an equation