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Mathematics · Grades 10–12

Functions: domains and transformations

Over real numbers, denominators cannot be zero and even-root inputs must be nonnegative.

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Understand the idea

Domain restrictions

Over real numbers, denominators cannot be zero and even-root inputs must be nonnegative.

Function notation

f(x) denotes an output, not f multiplied by x.

Examples and reasoning

For f(x) = 1/(x − 4), which input is excluded?

4

4

For f(x) = √(x − 3), the real domain is

x ≥ 3

x ≥ 3

Check your understanding

  1. The graph y = (x − 2)² is y = x² shifted

    1. 2 units right
    2. 2 units left
    3. 2 units down
    4. 2 units up

  2. The graph y = x² + 5 is y = x² shifted

    1. 5 units up
    2. 5 units right
    3. 5 units left
    4. 5 units down

  3. If f(x) = 2x + 1 and g(x) = x², find f(g(3)).

  4. An inverse function over a specified domain requires the original function to be

    1. one-to-one on that domain
    2. always positive
    3. always a straight line
    4. defined at every real number

Answers and marking notes

Answers and marking notes
  1. 2 units right

    2 units right

  2. 5 units up

    5 units up

  3. 19

    g(3) = 9; f(9) = 19.

  4. one-to-one on that domain

    one-to-one on that domain

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.

Learning preview · teacher review required. Grade bands are approximate, not official curriculum alignment.