Q.Let $\mathbb{R}$ be the set of real numbers. Define the real function $f : \mathbb{R} \to \mathbb{R}$ by $f(x) = x + 10$ and sketch the graph of this function.

Linear Function Evaluation — From Intuition to Precision

Imagine you're running a small stall that sells lemonade. Each glass costs ₹10. If you sell 1 glass, you earn ₹10. If you sell 5 glasses, you earn ₹50. If you sell x glasses, you earn ₹10x. That's a linear relationship — every extra glass adds exactly the same amount to your earnings.

This is the core idea behind a linear function: a rule that takes an input, multiplies it by a fixed number (the rate of change), and often adds a fixed starting amount.

The Intuition: A Machine That Adds the Same Amount Each Time

Think of a linear function as a simple machine. You feed it a number (the input), and it does two things in order:

1. Multiply that number by a constant (the slope or rate).
2. Add another constant (the intercept or starting value).

The output is the result. The key property: if you increase the input by 1, the output always increases by the same fixed amount (the slope). No surprises, no sudden jumps.

The Precise Statement

A linear function is any function that can be written in the form:

where:

• $x$ is the input (independent variable)
• $f(x)$ is the output (dependent variable)
• $m$ is the slope — the constant rate of change
• $b$ is the y-intercept — the output when $x = 0$

Evaluating a linear function means: given a specific input value, compute the corresponding output by substituting that value into the formula.

How to Evaluate — Step by Step

Suppose $f(x) = 3x + 5$. To evaluate $f(2)$:

1. Replace every $x$ in the formula with $2$:

2. Do the arithmetic:

That's it. The output when the input is $2$ equals $11$.

Why This Matters for Exams

Linear function evaluation is the foundation for:

• Solving linear equations (set $f(x) = \text{something}$ and solve for $x$)
• Understanding graphs (the slope tells you steepness, the intercept tells you where the line crosses the y-axis)
• Word problems (identify the rate and starting value, then write $f(x)$)

Common mistake to avoid: forgetting that $b$ is added after multiplication. If $f(x) = 2x + 3$, then $f(4) = 2(4) + 3 = 11$, not $2(4+3) = 14$.

A Quick Table of Examples

The Big Picture

Linear function evaluation is just plugging in a number and simplifying. The real power comes later — when you use this skill to model real situations, solve for unknowns, and understand how quantities change together. But for now, master the mechanics: replace $x$ with the given number, multiply by $m$, add $b$, and write the answer.

The final answer: Evaluating a linear function $f(x) = mx + b$ at $x = a$ means computing $f(a) = m \cdot a + b$.