Q.Let $f : \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = \begin{cases} 2x, & x > 3 \\ x^2, & 1 < x \leq 3 \\ 3x, & x \leq 1 \end{cases}$. Then $f(-1) + f(2) + f(4)$ is
(A) 9
(B) 14
(C) 5
(D) none of these

Piecewise function evaluation. For each input, first decide which interval it falls in, then apply only that branch's rule — never mix branches.

1. $x = -1$: since $-1 \leq 1$, the branch $f(x) = 3x$ applies, so $f(-1) = 3(-1) = -3$.
2. $x = 2$: since $1 < 2 \leq 3$, the branch $f(x) = x^2$ applies, so $f(2) = 2^2 = 4$. …

The key to piecewise functions is always the same: check the condition on the input before you apply the formula. Each piece of the definition has a domain interval, and you must pick the one that contains the given $x$-value. A common mistake is to use the wrong branch because you misread the inequality signs — especially the boundary points.

Let’s go through each term one at a time.

1. $f(-1)$ Look at the conditions:
   • $x > 3$? No, $-1$ is not greater than $3$.
   • $1 < x \leq 3$? No, $-1$ is not in $(1, 3]$.
   • $x \leq 1$? Yes, $-1 \leq 1$ is true. So we use the third branch: $f(x) = 3x$.

2. $f(2)$ Check:
   • $x > 3$? No.
   • $1 < x \leq 3$? Yes, $2$ lies between $1$ and $3$ (and note that $2$ is not equal to $1$, so the strict inequality on the left is fine). So we use the second branch: $f(x) = x^2$.

3. $f(4)$ …

Method: Evaluating a Piecewise Function at Given Points

Use this whenever a function is defined by different rules on different intervals and you must find its value at specific inputs.

Steps

Step 1: For each input, test the conditions in turn.

Compare the input against each interval's inequality to find the single branch it satisfies.

Step 2: Watch the boundary points and the strict vs non-strict signs. …

Common Mistakes

Mistake 1: Using the wrong branch by misreading the inequality.

Why it's wrong: each piece is valid only on its stated interval; using $2x$ for a value with $x\leq 1$ gives a wrong result. Correct approach: verify the input satisfies the branch's condition before substituting.

Mistake 2: Mishandling a boundary value like $x=1$ or $x=3$. …