# Function Notation Problems — Free Timed Practice | Quant Daily > Practise function notation problems free and timed: evaluate f(x), work inside-out on compositions, undo with inverses, and see a worked solution. No sign-up. Source: https://quantdaily.app/drill-types/function-notation --- Functions ## Function notation problems — substitute, compose, invert Function notation problems come in three shapes. Evaluating f(3) means substituting 3 wherever x appears. A composition such as f(g(2)) is worked from the inside outward, applying g first. An inverse asks the reverse question: given the output, recover the input, which you solve by setting the rule equal to that output. The notation does all the confusing work here. f(3) looks like multiplication and is not; f(g(2)) looks symmetric and is not. Once you read f(3) as 'put 3 into the rule' and f inverse of 14 as 'which input gives 14', the arithmetic that remains is a single line, and the whole topic becomes some of the fastest marks available. **Try it against the clock.** Functions runs at 90s easy, 105s medium and 120s hard. The clock is stamped and judged on our server, so the limit you see is the deadline that is actually enforced — and it starts when you tap Start, not while the question is loading. [Start a timed Functions drill →](https://quantdaily.app/?drill=functions_eval&src=seo-function-notation) ### How to work function notation problems in three shapes - Substitute the value everywhere x appears With f(x) = 5x + 1, f(3) is 5 times 3 plus 1, which is 16. The notation is an instruction to substitute, never a product of f and 3. - Compose from the inside out For f(g(2)), evaluate g at 2 first, then feed that result into f. The order matters: f(g(x)) and g(f(x)) are different functions and both answers are offered. - Invert by solving for the input With f(x) = 3x + 2, finding f inverse of 14 means solving 3x + 2 = 14, so 3x = 12 and x = 4. You are recovering the input, not evaluating the rule. - Check an inverse by substituting back If f inverse of 14 is 4, then f(4) should be 14. This one-line check catches nearly every arithmetic slip and costs a couple of seconds. ### What each piece of notation is asking for | Notation | Means | | --- | --- | | f(3) | Substitute 3 for x in the rule | | f(g(2)) | Apply g first, then f to that result | | g(f(2)) | Apply f first — a different answer | | f⁻¹(14) | Which input gives an output of 14 | | f(f(2)) | Apply f twice, inside out | ### 3 sample function notation problems, with worked solutions Generated by the same engine that mints the Quant Daily daily. Each answer is computed from the numbers printed in the question, and the walkthrough below each one is the engine's own working — not a solution written afterwards. Reload this page's live drill and you get different numbers. #### Question 1 f(x) = 4x + 4 What is f(3)? - 16 - 12 - 7 - 8 - 36 - f(3) = 4(3) + 4 = 16. - Answer: 16. #### Question 2 f(x) = 4x What is f(1)? - 4 - 8 - 6 - 1 - 2 - f(1) = 4(1) + 0 = 4. - Answer: 4. #### Question 3 f(x) = 2x + 4 What is f⁻¹(6)? - 6 - 5 - 2 - 1 - 4 - y = 2x + 4 ⇒ 2x = 2 ⇒ x = 1. - Check: f(1) = 6. - Answer: 1. ### Common mistakes this drill is built from The wrong options are not random numbers. Each one is the result of a specific careless error for this topic, so picking one tells you which habit is costing you marks. #### Reading f(3) as f times 3 The parentheses signal substitution, not multiplication. Treating them as a product produces a number that is always among the options. #### Composing in the wrong order In f(g(2)) the inner function runs first. Applying f first answers a different question, and that answer is printed too. #### Evaluating the rule when the inverse was asked for f inverse of 14 is not f(14). One asks which input gives 14; the other puts 14 in. Both are on the option list. ### Why your answer came out wrong The specific wrong numbers this topic produces, and what each one tells you about the step you took. If you have just got a question wrong and want to know which habit did it, start here. #### With f(x) = 3x + 2, is f⁻¹(14) equal to 44? 44 is f(14), which is the opposite operation. The inverse asks which input yields 14, so solve 3x + 2 = 14 to get x = 4. Check it by substituting back: f(4) is indeed 14. Both 44 and 4 appear among the options. #### Does f(g(2)) give the same answer as g(f(2))? Only by coincidence. Composition is not commutative, so the order of application changes the result in almost every case. The inner function is evaluated first, and questions routinely print both values so that a reversed order still finds something to select. ### FAQ #### How do I evaluate a composition like f(g(2))? Work from the inside out. Evaluate g at 2 first, then substitute that result into f. The brackets tell you the order, and reversing it gives g(f(2)), which is a genuinely different value that is usually offered as a distractor. #### What does f⁻¹(14) actually ask? It asks which input produces an output of 14. With f(x) = 3x + 2 you set 3x + 2 = 14 and solve, giving x = 4. It is the reverse of evaluating, so substituting 14 into the rule answers the wrong question entirely. #### Is f(x) the same as f times x? No. The parentheses in function notation mean 'the value of f at x', which is an instruction to substitute. This is one of the few places in algebra where brackets do not indicate multiplication, and it is the source of most early errors with the notation. ### Practise the next topic - [All 51 drill types](https://quantdaily.app/drill-types) — the shared daily cycles 36 of 51, with every published time limit. - [Systems of equations problems — eliminate](https://quantdaily.app/drill-types/systems-of-equations) — systems of equations problems, timed. - [Arithmetic and geometric sequences — find the rule](https://quantdaily.app/drill-types/sequences-ap-gp) — arithmetic and geometric sequences, timed. - [Speed tricks](https://quantdaily.app/drill-types) — the seven mental-math shortcuts that make a 15-second answer possible. - [How to improve calculation speed](https://quantdaily.app/calculation-speed) — the six-week arc these drills fit into. Quant Daily is free, needs no account, and is not affiliated with ETS or GMAC. GRE/GMAT-style practice only.