Given f(x) = -x - 4 for x < 3, f(x) = x squared - 7 for 3 <= x <= 10, and f(x) = 120/x + 5 for x > 10, find f(4).

Step-by-step solution

1. 1

   Understand what a piecewise definition asks of you. Three formulas are given, but only one applies at any given input. The first job is never to compute - it is to decide which condition the input satisfies.

2. 2

   Test x = 4 against each condition in turn. Is $4<3$? No. Is $3 \le 4 \le 10$? Yes. Is $4>10$? No. Exactly one condition holds, as it must for $f$ to be a well-defined function.

3. 3

   Select the matching rule. The middle condition applies, so on this interval $f(x)=x^2-7.$ The other two formulas are irrelevant here and must not be averaged or combined.

4. 4

   Substitute and evaluate. $f(4) = 4^2-7 = 16-7 = 9.$

5. 5

   Guard against the common mistakes. Using the first branch would give $-4-4=-8$ and the third $\tfrac{120}{4}+5=35$ - both wrong, because $4$ does not satisfy their conditions. Note also that the branch boundaries $3$ and $10$ are included in the middle piece, so $f(3)=2$ and $f(10)=93$ also come from $x^2-7$.