TMUA Even/Odd Functions and Symmetry

TMUA frequently creates questions which rely on the ability to spot even functions (f(x)=f(-x)) and odd functions (f(x)=-f(-x)), as well as functions symmetrical about other lines, either specifically, or indirectly where a question is far easier to answer if the symmetry is spotted. These questions can be testing as diverse topics as functions, integration, differentiation or geometry.

For those so interested, we offer both one to one TMUA coaching on Even/Odd Functions and Symmetry, as well as online group coaching via Zoom - 'TMUA Functions (inc Modulus, Even and Odd)' is Session #10 of 20. Check our TMUA Group Coaching Schedule page for dates when this session is available - details of what we cover in each group session can be found on our TMUA Coaching Sessions Overview page. For more general information on our online sessions check out our  'Online Coaching FAQ'.

Feel free to contact us if you require more information regarding any of the above.

Free Revision Resources

The below list of questions links to short videos (teasers) on our YouTube channel which directly or indirectly test elements of what is required to succeed in answering TMUA questions on even/odd functions and symmetry - they also provide a detailed solution to the question posed, together with tips and tricks and are arranged roughly in increasing order of difficulty. All the videos are taken from the Gresty Academy YouTube podcast 'A Crash Course in TMUA Must Know Maths' which contains hundreds of TMUA style questions and is great for revision.

• Teaser #423 - tests how to find the number of perfect square factors (ie products of even powers) of 162,000 - Standard
• Teaser #103 - tests understanding of symmetry of a quadratic - Standard
• Teaser #19 - tests finding equation of a reflection of a parabola in the y-axis - Standard
• Teaser #466 - tests ability to spot the type of symmetry of a functional equation - Standard
• Teaser #449 - tests how to find A when sin^2(Ax) satisfies f(x)=f(x+2) - Medium
• Teaser #447 - tests knowledge of finding point of rotational symmetry from a functional equation - Medium
• Teaser #446 - tests knowledge of which functional equations represent symmetry in the line x=a - Medium
• Teaser #430 - tests understanding of how to find point of rotational symmetry - Medium
• Teaser #400 - tests if you know a quick method to find the sum of gradients of an even function - Medium
• Teaser #275 - tests whether composites of even and odd functions are even, odd or neither - Medium
• Teaser #26 - tests following sequences of transformations on a curve - Medium
• Teaser #487 - tests understanding of the integral of f(|x|) - Medium
• Teaser #204 - tests ability to solve an equation involving multiple 'tan' taking advantage of graphical symmetry - Medium
• Teaser #500 - tests understanding of when sqrt(x^2) = (sqrt(x))^2 - Medium
• Teaser #171 - tests ability to solve an equation involving multiple 'sin' taking advantage of graphical symmetry - Medium
• Teaser #405 - tests ability to find the area enclosed between an even function and the x-axis - Medium
• Teaser #484 - tests ability to use symmetry and patterns to sum a tricky looking trigonometric series - Medium
• Teaser #486 - tests recognition of a well known functional inequality - Medium
• Teaser #172 - tests how to find the area of a square inside multiple circles using symmetry - Medium
• Teaser #203 - tests how to find the radius of a circle inscribed in an equilateral triangle using symmetry - Medium
• Teaser #502 - tests ability to split a function into its odd and even components - Medium
• Teaser #280 - tests knowledge of how to graph the modulus of a quadratic, and find the product of roots using the fact the amended function is even - Medium
• Teaser #310 - tests knowledge of how to graph the modulus of a quadratic, and find the roots using the fact the amended function is even - Medium 
• Teaser #479 - tests understanding of which functional equations are sufficient for an integral to hold true - Quite Tough
• Teaser #467 - tests ability to execute transformations in the right order and then recognise a functional equation - Quite Tough
• Teaser #420 - tests ability to find number of solutions to x^2 sin(2x)=cos(2x) - Quite Tough
• Teaser #489 - tests ability to integrate a product of two modulus - Quite Tough
• Teaser #492 - tests ability to use graphical methods and a symmetry trick to solve a tough looking modulus equation - Quite Tough
• Teaser #461 - tests ability to establish which functional equations have solutions and which don't - Quite Tough
• Teaser #262 - tests noticing even and odd functionality to massively simplify an integral - Quite Tough
• Teaser #480 - tests ability to split a function into its odd and even components to solve a functional equation - Quite Tough
• Teaser #282 - tests noticing even and odd functionality to massively simplify an integral - Quite Tough

[Note: We also have a 'Functions' playlist on our YouTube channel however the videos in this playlist are not all relevant to TMUA (some are too easy, some are too hard, and many cover topics not tested in TMUA).]

TMUA Past Questions on Even/Odd Functions and Symmetry: SP p2 q19; 2016 p2 q11; 2016 p2 q18; 2017 p1 q15; 2017 p2 q10; 2018 p1 q18; 2018 p2 q10; 2018 p2 q15; 2019 p1 q8; 2019 p1 q19; 2019 p2 q20; 2020 p1 q5; 2020 p1 q14; 2020 p2 q6; 2021 p1 q7; 2022 p1 q7; 2022 p1 q14; 2023 p2 q20

Practice makes perfect - good luck in your studies!