Decoding f(3): A Simple Step-by-Step Evaluation of f(x) = 3 − 3²/2

Understanding how to evaluate mathematical expressions is a foundational skill in algebra and beyond. Today, we break down a specific function: f(3) = 3 − 3²⁄2, and walk through every step to arrive at the final answer of −1.5. This process not only clarifies a single problem but also strengthens key algebraic reasoning skills.

Understanding the Context

Breaking Down the Expression f(3) = 3 − 3²⁄2

At first glance, the expression
f(3) = 3 − 3²⁄2
may look complex due to the combination of constant, exponentiation, and rational division. However, by following standard order-of-operations rules (PEMDAS/BODMAS), we simplify it step by step.

Step 1: Exponentiation — Evaluate 3²

$=\frac{x}{2}+\frac{1}{2} \log |\cos x+\sin x|+\frac{9}{2}+\frac{3}{2}$

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$=\frac{x}{2}+\frac{1}{2} \log |\cos x+\sin x|+\frac{9}{2}+\frac{3}{2}$

Ex 2.3, 3 - f(x) = 2x - 5. Write down f(0), f(7), f(-3) - Class 11

Key Insights

In algebra, the symbol 3² means 3 squared, which equals 3 × 3 = 9.
So,
3² = 9

Now substitute back:
f(3) = 3 − 9⁄2

Step 2: Division — Evaluate 9⁄2

The expression now contains a division: 9 ⁄ 2.
This is a proper fraction, representing 9 divided by 2, which equals 4.5 in decimal form.
So,
9⁄2 = 4.5

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Final Thoughts

Now the function becomes:
f(3) = 3 − 4.5

Step 3: Subtraction — Final Computation

Subtracting a larger number from a smaller one gives a negative result:
3 − 4.5 = −1.5

Thus,
f(3) = −1.5

Why Understanding f(3) Matters

While this function is simple, mastering such evaluations builds confidence in handling linear, quadratic, and rational expressions. Functions like f(x) form the backbone of higher mathematics—from calculus to computer science.

Understanding each operation’s place in the calculation chain helps prevent errors, especially when solving equations, analyzing graphs, or optimizing real-world problems.