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CAT Functions questions with solutions

Function evaluation and functional equations. 22 questions from our 2016–2025 CAT pattern papers. Each has the answer and a full solution.

CAT 2025

A function is defined as f(x) = 2x + 3 for x < 0, and f(x) = x² − 1 for x ≥ 0. What is the value of f(f(−2))?CAT 2025 Slot 1 · hard
A function f satisfies f(x+y) = f(x) + f(y) + xy for all real x, y, and f(1) = 2. Find f(10).CAT 2025 Slot 2 · hard
For x ≠ 2, define f(x) = (x+3)/(x−2). Other than x = 2, for what value of x is f(f(x)) undefined?CAT 2025 Slot 3 · hard

CAT 2024

A function f satisfies f(x) + 3f(4−x) = 2x + 1 for all real x. Find f(2).CAT 2024 Slot 1 · medium
A function f satisfies f(x+y) = f(x)·f(y) for all real x, y, and f(1) = 3. Find f(5).CAT 2024 Slot 3 · hard

CAT 2023

A function g satisfies g(x) = 2·g(x−1) − 1 for all integers x, with g(1) = 3. Find g(6).CAT 2023 Slot 1 · hard
A function f satisfies f(x) + 2f(6−x) = 3x for all real x. Find f(2).CAT 2023 Slot 2 · hard

CAT 2022

A function is defined as f(x) = 2x+3 for x ≤ 2, and f(x) = x²−1 for x > 2. Find the value of f(f(2)).CAT 2022 Slot 2 · medium
A function f satisfies f(x) + 3f(4 − x) = 2x for all real x. Find f(1).CAT 2022 Slot 3 · hard

CAT 2021

A function f, defined for all non-zero real x, satisfies f(x) + 2f(1/x) = 3x. What is the value of f(2)?CAT 2021 Slot 1 · hard
A function f, defined for all non-zero real x, satisfies f(x) + 2f(1/x) = 3x. For how many non-zero real numbers x is f(x) = f(−x)?CAT 2021 Slot 2 · hard
A function f, defined for all non-zero real numbers, satisfies f(x) + 2f(1/x) = 3x for every x ≠ 0. What is the sum of all real values of x for which…CAT 2021 Slot 3 · hard

CAT 2020

A function g satisfies g(x) + 3g(1/x) = 4x − 4 for every non-zero real x. What is g(3)?CAT 2020 Slot 1 · hard
If f(x) = (2x + 3)/(x − 2) for x ≠ 2, then f(f(f(5))) equalsCAT 2020 Slot 2 · medium
A function h satisfies h(x) + 4h(1/x) = 3x + 5 for every non-zero real x. What is h(2)?CAT 2020 Slot 3 · medium

CAT 2019

What is the value of f(2)?CAT 2019 Slot 1 · hard
A function f(x) = ax + b, with a > 0, satisfies f(f(x)) = 9x + 8 for all real x. What is f(3)?CAT 2019 Slot 2 · medium

CAT 2018

Find f(2).CAT 2018 Slot 1 · medium
What is f(3)?CAT 2018 Slot 2 · hard

CAT 2017

F(x) is a quadratic polynomial such that f(x + 1) + f(x − 1) = 2x² − 4x + 6 for all real x. What is f(3)?CAT 2017 Slot 2 · hard

CAT 2016

What is f(2)?CAT 2016 Slot 1 · hard
If f(x) = (x + 1)/(x − 1) for x ≠ 1, then f(f(f(3))) equals:CAT 2016 Slot 2 · medium