Step-by-Step Strategies for Functions and Inequalities in CAT Quant Section

Understand the Question

Take note that the domain of the function is the valid input while the range is the possible output

Check Domain of Function

In order to check domain restrict the value to

• Denominators (≠ 0)
• Square roots (≥ 0)
• Logarithms (> 0)
• Absolute values (case-split if needed)

Next Find Range

To find range follow the transformation method by using Quadratic, Rational and Composition techniques

Use Graphical Representation

Rough sketches help with symmetry (odd/even functions), increasing/decreasing behavior, and boundary values.

Simplify Expression

Factorise the quadratic equation. Bring the values to one side

Use Number Line Method

• Plot critical points on a line.
• Test sign in each interval.
• Decide open/closed interval depending on ≥, ≤

Absolute Value

Split into cases based on sign

Exponential / Logarithmic

• If base > 1 → inequality direction unchanged.
• If 0 < base < 1 → inequality flips.

Substitution Method

• When asked to find f(f(x)) or f(g(x)) substitute step by step

Tip:

If f(x) = x2+1 then f(f(x)) = f(x2+1) = (x2+1)2+1.

Use Graph Visualization

Replace f(x)=∣g(x)∣ with cases

• If g(x) ≥ 0 then f(x) = g(x)
• If g(x) < 0 then f(x) = - g(x)

Piecewise Functions

For modulus or floor functions, break into intervals

Example: f(x) = ∣x−2∣

Case 1 - x ≥ 2 → f(x) = x−2

Case 2 - x < 2 → f(x) = - (x - 2) = 2 - x

Tricks for Linear Inequalities

Solve like equations, but flip the inequality sign when multiplying/dividing by a negative number.

For example-

-2x > 6 ⟹ x < –3

Tricks for Quadratic Inequalities

• Factorize the quadratic (if possible).
• Find the critical points (roots)
• If coefficient of x² > 0 → parabola opens upward
• If the coefficient of x² < 0 → parabola opens downward.

Tricks for Rational Inequalities

• Factorize numerator & denominator separately.
• Mark critical points (where numerator = 0 or denominator = 0).
• Draw number line → test signs of each interval.
• Exclude points where denominator = 0.