Q.Show that the function given by $f(x) = 7x - 3$ is increasing on $\mathbb{R}$.

The key idea is the Increasing Function Test: a function is increasing on an interval if its derivative is non-negative at every point in that interval.

Step 1: Differentiate $f(x) = 7x - 3$ with respect to $x$:

Step 2: Since $7 > 0$, we have $f'(x) > 0$ for all $x \in \mathbb{R}$.

Step 3: A positive derivative everywhere implies the function is strictly increasing on $\mathbb{R}$.

The idea is simple: an increasing function always moves upward as you go right. For a differentiable function, this is captured by the Increasing Function Test — if the derivative is positive (or at least non-negative) at every point, the function cannot dip down, so it must be increasing.

Here’s why this works: the derivative $f'(x)$ measures the slope of the tangent line. A positive slope means the function is rising as $x$ increases — like walking uphill. If the slope is always positive, you never walk downhill, so the function never decreases.

Now let’s apply it to $f(x) = 7x - 3$.

1. Find the derivative. $f(x) = 7x - 3$ is a linear function. Differentiating term by term:

2. Check the sign of the derivative.

   $f'(x) = 7$ is a constant — it doesn’t depend on $x$. And $7 > 0$ for every real number $x$.

3. Apply the Increasing Function Test.

   Since $f'(x) > 0$ for all $x \in \mathbb{R}$, the function is strictly increasing on the entire real line.

Method: Proving a Function Is Increasing (or Decreasing) on an Entire Interval

Use this whenever a question asks you to show or prove that a given function is increasing (or decreasing) throughout an interval — as opposed to finding the intervals of increase/decrease, which needs a sign chart (see the Monotonic Function Analysis method).

Steps

Step 1: Differentiate the function.

Find $f'(x)$ using the standard differentiation rules.

Step 2: Argue that $f'(x)$ keeps one sign across the whole interval.

To prove a function increasing everywhere on an interval, you must show $f'(x) > 0$ (or $\ge 0$) for every $x$ in that interval — not just at one sample point. If $f'(x)$ turns out to be a positive constant, this is immediate; if it depends on $x$, you may need an algebraic argument (a perfect square, a sum of positive terms, or a discriminant check) to guarantee the sign never flips.

Step 3: Apply the Increasing/Decreasing Function Test.

Quote this theorem explicitly as the justification — a "show that" question expects the logical link between the derivative's sign and the monotonicity conclusion to be stated, not just the arithmetic.

Step 4: Conclude for the given interval.

State the interval (here $\mathbb{R}$, but the same logic restricts to any given domain) and the final conclusion.

Common Mistakes

Mistake 1: Confusing "increasing" with "the function's values are positive."

Why it's wrong: increasing is a statement about the slope — whether $f(x)$ rises as $x$ increases — not about whether $f(x)$ itself is a positive number. The function $f(x) = 7x - 3$ is negative for $x < 3/7$ but is still strictly increasing throughout, because its derivative is a positive constant everywhere. Correct approach: judge monotonicity purely from the sign of $f'(x)$, never from the sign of $f(x)$ itself.