0580

Functions

Algebra and Sequences

What each word covers

  • The domain is the collection of inputs a function is allowed to take
  • The range is the collection of outputs those inputs produce
Two functions and the graphs of y equals each: f(x) = 2x + 5 is linear, so its graph is a straight line crossing the y-axis at 5; g(x) = x² − 7x + 10 is quadratic, so its graph is a u-shaped curve crossing the x-axis at 2 and 5 and the y-axis at 10
Source: Domain & range by Save My Exams
  • A domain is always described in terms of x, and a range always in terms of f(x), and swapping the two loses the mark
  • On this specification the domain is very often handed to you as a short list of numbers inside curly brackets, and the range is then simply the list of answers you get by substituting each one
  • The outputs are listed in the same curly-bracket form, and getting most of them right still scores
Worked example

A range from a listed domain

g(x) = x² + 2 has domain {−3, 1, 4}. Find the range of g(x).

Solution:

  • The domain is the set of inputs allowed, and the range is the set of outputs they produce
  • The domain here is a short list, so the range is a short list too: substitute each value in turn
  • g(−3) = (−3)² + 2 — bracket the negative, then square it
  • (−3)² = 9, so g(−3) = 9 + 2 = 11
  • g(1) = 1² + 2 = 1 + 2 = 3
  • g(4) = 4² + 2 = 16 + 2 = 18
  • The range is {3, 11, 18}
  • Give the outputs, not the inputs: {−3, 1, 4} is the domain and answers a different question
  • Writing them in increasing order is conventional, and it also makes a missing value obvious

Domains given as an interval, and values that must be excluded

  • Where the domain is an interval such as 1 ⩽ x < 5, substitute each end value to find where the outputs start and stop
  • Carry the ⩽ and < across carefully, remembering that a decreasing function swaps which end is included
  • Some values can never be inputs, and the two reasons are dividing by zero and square rooting a negative
    • h(x) = 5 ÷ (x − 6) excludes x = 6, because the bottom would be zero
    • j(x) = √(x − 2) excludes everything below 2, because the inside would be negative
  • To find an excluded value, set the bottom of the fraction equal to zero and solve
  • A quick sketch of the graph settles most range questions faster than substituting does
A sketch of y = g(x) where g(x) = 3x² with domain x greater than or equal to 0: only the right-hand half of the u-shaped curve is drawn, rising from the origin, because the restricted domain keeps just part of the full graph
Source: Domain & range by Save My Exams
  • Composite functions are the one exception here: this specification does not ask for their domains or ranges
Worked example

A range from an interval

f(x) = 4x − 7 has domain 1 ⩽ x < 5. Find the range of f(x).

Solution:

  • With an interval there are too many inputs to list, so work from the ends
  • The rule multiplies by 4, which is positive, so larger inputs give larger outputs and the order of the ends is preserved
  • At the lower end: f(1) = 4 − 7 = −3
  • At the upper end: f(5) = 20 − 7 = 13
  • Each end keeps the symbol it had: 1 was included, so −3 is included; 5 was excluded, so 13 is excluded
  • The range is −3 ⩽ f(x) < 13
  • Write the range in terms of f(x), not x: −3 ⩽ x < 13 describes a set of inputs
  • Had the rule multiplied by a negative number the ends would swap over, so the check is always to substitute both and see which output is larger