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Mathematics · Grades 9–12

Functions, domains and inverse reasoning

A function assigns exactly one output to each allowed input.

Make a worksheet

Understand the idea

One input, one output

A function assigns exactly one output to each allowed input.

Restrictions matter

A formula may be undefined for some values.

Undo in reverse order

An inverse reverses a one-to-one function on its specified domain.

Worked examples

Which set of ordered pairs does not define y as a function of x?

Input 1 is assigned two different outputs.

{(1,2),(1,4)}

For f(x) = 3x − 7, calculate f(5).

3 × 5 − 7 = 8.

8

Try it yourself

  1. Which input must be excluded from g(x) = 1/(x − 6)?

    1. 0
    2. 6
    3. −6
    4. 1

  2. If f(x) = 2x + 9 and f(a) = 1, find a.

  3. Which is the inverse of f(x) = 5x − 2?

    1. f⁻¹(x) = x/5 − 2
    2. f⁻¹(x) = (x + 2)/5
    3. f⁻¹(x) = 1/(5x − 2)
    4. f⁻¹(x) = 5x + 2

  4. Why does x² on all real numbers not have an inverse function with all nonnegative outputs as inputs?

Answers and marking notes

Answers and marking notes
  1. 6

    At x = 6, the denominator is zero.

  2. -4

    2a + 9 = 1 gives 2a = −8.

  3. f⁻¹(x) = (x + 2)/5

    Solve y = 5x − 2 for x, then rename the variables.

  4. Both x and −x can give the same square. An inverse would not have a unique output unless the domain is restricted.

    For example, restrict the original domain to x ≥ 0.

Further reading and review references

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