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.

- y = 4x + 8

Domain : {all real x}

Range: {all real y}

This is a linear function. - y = | 2x + 5 |

Domain : {all real x}

Range: {y: y ≥ 0}

This is an absolute function, with a V-shaped curve. - y = f(x) = 4x
^{2}– 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). - 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. - y = f(x) = x
^{3}– 64

Domain : {all real x}

Range: {all real y}

This is a cubic function. - x
^{2}+ y^{2}= 25

Domain : {x: -5 ≤ x ≤ 5}

Range: {y: -5 ≤ x ≤ 5}

This is a circle. A circle is not a function. - 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) - 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** - g(t) = 2
^{t}

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. - 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. - 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} - 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: {}