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Domains and Ranges

Domains and Ranges

A domain defines a set of input to a function. The range is defined as the output.

For a linear function, such as `f(x) = 3x + 4`, the domain could be used for any Real (ℝ) number. The result - the range - would also be any real number and there would be only one output value for each input value.

A function must work for every possible input value, and for every input value there must be one output value. However, the output values can be shared: `f(x) = x^2` is a valid function, although `f(2)` and `f(-2)` give the same result (4).

The domain may have to be limited for certain functions. `f(x) = frac(1)(x)` works for any real number except the value 0, as an attempt to divide by zero will fail. Therefore `f(x) = frac(1)(x), x ≠0 ` is a valid function as the range has ben defined to eliminate the value 0.

The trigonometric ratios (sin, cos and tan) are all functions. Because the values repeat as you go above 360º, the function will be defined with a limiting range eg `f(x) = sin(x), 0º ≤ x < 360º` to avoid the repeating values.

Sometimes the domain or range will be defined as a set (using set notation { } ); and sometimes using one of the predefined numeric types: the integer numbers shown as ℤ; rational numbers shown as ℚ, Real numbers shown as ℝ, and prime numbers.

Example 1

What is the range for the function `f(x) = x^2 - 4`, where 3 ≤ x ≤ 5?

`f(3) = 3^2 - 4 = 5`

`f(5) = 5^2 - 4 = 21`

`5 ≤ f(x) ≤ 21`

Answer: `5 ≤ f(x) ≤ 21`

Example 2

For `f(x) = x^2 - 4`, the domain is `a ≤ x ≤ 8`. If the range is `5 ≤ f(x) ≤ 60`, what is the value of `a`?

The lowest value:

`5 = x^2 - 4`

`x^2 = 9`

`x = 3`

Answer: `a = 3`

See also Sets in Practical Situations