Functions, Domains, Ranges and Optimization Short Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Functions, Domains, Ranges and Optimization short notes for CAT: 25 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Composition Means: Inner Function First

Composition flow

Summary

g (x)

then

f (g (x))

Always start with the innermost expression.

Functional Equations: Create a Matching Equation

When $f (x)$ is linked with another input

If you see:

f (x) + 2 f (\frac{1}{x}) = 3 x

also write the equation after replacing

x

\frac{1}{x}

Step 1: Identify the paired input.

Step 2: Write the second equation using that input.

Step 3: Solve the two equations for the needed function value.

Pattern 1: Function of Two Linear Expressions

Reverse the input transformation

PYQ Pattern

If:

u = 3 x + 2 y, v = 2 x - 5 y

find

x

in terms of

u, v

Here:

5 u + 2 v = 5 (3 x + 2 y) + 2 (2 x - 5 y) = 19 x

So a scary two-variable function can become a simple linear rule.

Piecewise Functions: First Decide the Case

Piecewise means rule depends on input

f (x) = {2 x - r, r, x \geq r x < r

If input is at least

r

, use the first rule.

If input is less than

r

, use the second rule.

For

f (f (x))

, first find

f (x)

, then decide which case that output belongs to.

Functions, Domains, Ranges and Optimization: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Quantitative Ability NAT

Let $f (x) = \frac{x ^{2} - x + 1}{x ^{2} + x + 1}$ for all real $x$ . If $M$ and $m$ are the maximum and minimum values of $f (x)$ respectively, what is the value of $3 (M + m)$ ?

Correct Answer:

Step-by-Step Solution

Key idea: This is an optimization via discriminant question. When finding the range (max/min values) of a rational function where both numerator and denominator are quadratics, the most efficient method is to use the discriminant.

Step 1: Set the function equal to $y$ .

$y = \frac{x ^{2} - x + 1}{x ^{2} + x + 1}$ .

Step 2: Cross-multiply and rearrange into a standard quadratic equation in terms of $x$ .

$y (x^{2} + x + 1) = x^{2} - x + 1$

$y x^{2} + y x + y = x^{2} - x + 1$

$(y - 1) x^{2} + (y + 1) x + (y - 1) = 0$ .

Step 3: Apply the condition for real roots.

Since the domain is all real $x$ , for any valid output $y$ , there must exist at least one real $x$ that satisfies this equation. Therefore, the discriminant $Δ$ must be greater than or equal to zero.

$Δ = b^{2} - 4 a c \geq 0$

$(y + 1)^{2} - 4 (y - 1) (y - 1) \geq 0$ .

Step 4: Solve the inequality for $y$ .

Expand the terms:

$(y^{2} + 2 y + 1) - 4 (y^{2} - 2 y + 1) \geq 0$

$y^{2} + 2 y + 1 - 4 y^{2} + 8 y - 4 \geq 0$

$- 3 y^{2} + 10 y - 3 \geq 0$ .

Multiply by $- 1$ and flip the inequality sign:

$3 y^{2} - 10 y + 3 \leq 0$ .

Factor the quadratic:

$(3 y - 1) (y - 3) \leq 0$ .

The roots are $y = 1/3$ and $y = 3$ . Since the parabola opens upwards and is $\leq 0$ , the solution is the interval between the roots:

$1/3 \leq y \leq 3$ .

Step 5: Identify $M$ and $m$ and calculate the final answer.

The maximum value $M = 3$ .

The minimum value $m = 1/3$ .

The question asks for $3 (M + m)$ :

$3 (3 + 1/3) = 3 (10/3) = 10$ .

Answer: 10

Question 2 · Quantitative Ability NAT

How many integer values are in the range of the function $f (x) = \frac{8 x - 16}{x ^{2} - 4 x + 5}$ for all real numbers $x$ ?

Correct Answer:

Step-by-Step Solution

Key idea: This is a linear over quadratic range question. The trigger is a rational function where the numerator is degree 1 and the denominator is degree 2. The most robust method is substitution to simplify the expression, followed by the discriminant method to find the exact bounds of the range.

Step 1: Simplify the function using substitution.

Notice the denominator: $x^{2} - 4 x + 5 = (x^{2} - 4 x + 4) + 1 = (x - 2)^{2} + 1$ .

Notice the numerator: $8 x - 16 = 8 (x - 2)$ .

Let $u = x - 2$ . As $x$ ranges over all real numbers, $u$ also ranges over all real numbers.

The function becomes $y = \frac{8 u}{u ^{2} + 1}$ .

Step 2: Set up the quadratic equation in terms of $u$ .

Cross-multiply to get $y (u^{2} + 1) = 8 u$ .

$y u^{2} - 8 u + y = 0$ .

Step 3: Apply the discriminant condition for real $u$ .

Case A: If $y = 0$ , the equation becomes $- 8 u = 0 ⟹ u = 0$ . Since $u = 0$ is a real number, $y = 0$ is in the range.

Case B: If $y \neq = 0$ , this is a quadratic equation in $u$ . For real solutions to exist, the discriminant $D$ must be non-negative.

$D = B^{2} - 4 A C = (- 8)^{2} - 4 (y) (y) \geq 0$ .

$64 - 4 y^{2} \geq 0$ .

$4 y^{2} \leq 64 ⟹ y^{2} \leq 16$ .

Taking the square root gives $- 4 \leq y \leq 4$ .

Step 4: Combine and count the integers.

The full range is the closed interval $[- 4, 4]$ .

The integer values in this interval are: $- 4, - 3, - 2, - 1, 0, 1, 2, 3, 4$ .

Counting these gives exactly 9 integers.

Answer: 9

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Functions, Domains, Ranges and Optimization Short Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Functions, Domains, Ranges and Optimization short notes for CAT: 25 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice qu

A question from this chapter

Question 1

Let f (x) = \frac{x ^{2} - x + 1}{x ^{2} + x + 1} for all real x . If M and m are the maximum and minimum values of f (x) respectively, what is the value of 3 (M + m) ?

Question 2

How many integer values are in the range of the function f (x) = \frac{8 x - 16}{x ^{2} - 4 x + 5} for all real numbers x ?

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Functions, Domains, Ranges and Optimization Short Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Composition Means: Inner Function First

Composition flow

Functional Equations: Create a Matching Equation

When f(x) is linked with another input

Pattern 1: Function of Two Linear Expressions

Reverse the input transformation

Piecewise Functions: First Decide the Case

Piecewise means rule depends on input

Functions, Domains, Ranges and Optimization: Solved Questions with Step-by-Step Explanations (2 Problems)

Step-by-Step Solution

Step-by-Step Solution

More short notes in this unit

A question from this chapter

Login to view the complete short notes

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

Revision that hits your weak spots

Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course

When $f (x)$ is linked with another input