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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.

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How to work function notation problems in three shapes

  1. 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.

  2. 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.

  3. 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.

  4. 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

NotationMeans
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)?

  1. 16
  2. 12
  3. 7
  4. 8
  5. 36
Show the worked solution
  1. f(3) = 4(3) + 4 = 16.
  2. Answer: 16.

Question 2

f(x) = 4x

What is f(1)?

  1. 4
  2. 8
  3. 6
  4. 1
  5. 2
Show the worked solution
  1. f(1) = 4(1) + 0 = 4.
  2. Answer: 4.

Question 3

f(x) = 2x + 4

What is f⁻¹(6)?

  1. 6
  2. 5
  3. 2
  4. 1
  5. 4
Show the worked solution
  1. y = 2x + 4 ⇒ 2x = 2 ⇒ x = 1.
  2. Check: f(1) = 6.
  3. 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.

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