Question

Consider the piecewise function \[ f(x) = \begin{cases} 3 - x, & x \leq 0 \\ x^2 + 2, & x > 0 \end{cases} \] Evaluate \(f(-3)\).

• A. 6
• B. 0
• C. 11
• D. -7

Hint:

Since \(-3 \leq 0\), use the branch \(f(x) = 3 - x\), not \(x^2+2\), which only applies for \(x > 0\).

Answer 1

The Correct Option is A

A piecewise function is defined by different formulas depending on which range the input falls into. Here, $f(x) = 3 - x$ applies when $x \leq 0$, and $f(x) = x^2 + 2$ applies when $x > 0$. Since we want $f(-3)$ and $-3$ is less than $0$, only the first rule can be used. Let's check what each option would require.

1. 6: substituting $x = -3$ into the valid branch $3 - x$ gives $3 - (-3) = 3 + 3 = 6$. This matches the branch that actually applies to $x = -3$, so it is correct.
2. 0: this would only happen from a sign error, computing $3 - 3$ instead of $3 - (-3)$. There is no legitimate way to reach 0 from either branch at $x=-3$.
3. 11: this comes from plugging $x = -3$ into the wrong branch, $x^2 + 2 = 9 + 2 = 11$. That branch is reserved for $x > 0$, and $-3$ fails that condition, so this option uses the wrong formula.
4. -7: neither branch produces this value when $x = -3$ is substituted correctly, so it is simply an incorrect distractor.

Because $-3 \leq 0$, the rule $f(x) = 3 - x$ is the one that governs this input, and it gives $f(-3) = 6$.

Let's summarize:

• Always check which condition the input satisfies before picking a branch.
• Only after choosing the right branch should you substitute and simplify.

So the value of $f(-3)$ is 6.