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Examples of Domain and Range

We now look at a few examples of domain and range for each type of function below – linear, absolute, parabola, hyperbolic, cubic, circle, exponential, top half of a circle, top half of a parabola, etc.

  1. y = 4x + 8
    Domain : {all real x}
    Range: {all real y}
    This is a linear function.
  2.  y = | 2x + 5 |
    Domain : {all real x}
    Range: {y: y ≥ 0}
    This is an absolute function, with a V-shaped curve.
  3.  y = f(x) = 4x2 – 1
    Domain : {all real x}
    Range:  {f(x): f(x) ≥ -1}  or {y: y ≥ -1}
    This is a parabola (parabola usually has a quadratic function).
  4.  y = f(x) =  3/{t~-~5}
    When t = 5, f(t) is undefined. So this function is discontinuous.
    Domain : {all real t: t ≠ 5}
    Range:  {all real f(t): f(t) ≠ 5}
    This is a hyperbolic function.
  5.  y = f(x) = x3 – 64
    Domain : {all real x}
    Range:  {all real y}
    This is a cubic function.
  6.  x2 + y2 = 25
    Domain : {x: -5 ≤ x ≤ 5}
    Range:  {y: -5 ≤ x ≤ 5}
    This is a circle. A circle is not a function.
  7.  y = sqrt{36~-~x^2}
    Domain : {x: -6 ≤ x ≤ 6}
    Range:  {y: 0 ≤ x ≤ 6}
    This is a function and represents the top half of a circle (positive semi-circle)
  8.  y = sqrt{x~-~4}
    Since we can find the square of a positive number only, x – 4 ≥ 0. So x ≥ 4.
    When we take the square root of a number, we consider only the positive or zero number. So y ≥ 0.Domain : {x: x ≥ 4}
    Range:  {y: y ≥ 0}
    This is a function and represents the top half of a parabola
  9.  g(t) = 2t
    Since we can find the square of a positive number only, x – 4 ≥ 0. So x ≥ 4.
    When t is very small, t = –infty, so g(t) = 2^{-infty}  =  1/2^{-infty}  right 0, but does not reach 0.
    Domain : {x: all real t}
    Range:  {g(t): g(t) >0}
    This is function represents an exponential function.
  10.  P (alpha) = log_a{alpha}
    We know log 1 = 0 and we can’t find a value for log 0 or any negative numbers.
    Hence when x>0, y exists for all x
    Domain : {All real x: x > 0}
    Range:  {All real y}
    This is a function representing a logarithmic function.
  11.  y = sqrt{x^2~-~x~-~2}
    =  sqrt({x~+~1)(x~-~2)}
    For y to exist, x² – x – 2 ≥ 0
    Since we can find the square of a positive number only, y is always zero or a positive value.
    Therefore Range: {y: y ≥ 0}
    Now x² – x – 2 ≥ 0 or (x – 2)(x + 1) ≥ 0
    Or x ≤ -1 and x ≥ 2
    Domain : {x: x ≤ -1 and x ≥ 2}
    Range:  {y: y ≥ 0}
  12.  y = f(x) =  {delim{|}x{|}}/x
    Here x~<>~0, so when x < 0: y = {-x}/x  =  -1
    when x > 0: y = x/x  =  1
    Therefore y is always -1 or 1.
    Domain : {all real x: x<>0}
    Range:  {{y~=~pm{1}}}