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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) =
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 =
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 =
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 = –, so g(t) =   =     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 () =
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 =
=
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) =
Here , so when x < 0: y =   =  -1
when x > 0: y =   =  1
Therefore y is always -1 or 1.
Domain : {all real x: }
Range:  {}