Understanding the Range in Mathematics: Methods and Examples for Finding the Set of Possible Output Values

range

In mathematics, the range refers to the set of all possible values that a function or relation can take

In mathematics, the range refers to the set of all possible values that a function or relation can take. It is the collection of all output values that correspond to the input values.

To find the range of a function or relation, you need to look at all the possible output values. Here are two methods to determine the range:

1. Graphical approach:
– Plot the graph of the function or relation.
– Look at the y-values or the vertical axis of the graph.
– Identify the highest and lowest y-values on the graph.
– The range is the set of all y-values between the highest and lowest values, including both endpoints.

2. Algebraic approach:
– Start with the function or relation equation.
– Solve the equation for all possible output values.
– Take note of any restrictions on the domain or possible inputs.
– The resulting set of output values is the range.

Let’s look at an example to illustrate the concept.
Consider the function f(x) = x^2, where x is a real number.

1. Graphical approach:
– Plot the points on a graph: (0, 0), (1, 1), (-1, 1), (2, 4), (-2, 4), etc.
– The graph forms a parabolic shape, opening upwards.
– The lowest y-value on the graph is 0, and there is no upper bound.
– Therefore, the range is all real numbers greater than or equal to 0: Range = [0, ∞).

2. Algebraic approach:
– Start with f(x) = x^2.
– Since the square of any real number is always non-negative, there are no restrictions on the output values.
– Therefore, the range is again all real numbers greater than or equal to 0: Range = [0, ∞).

In summary, when finding the range of a function or relation, you need to consider all possible output values. This can be accomplished graphically by examining the y-values on a graph or algebraically by solving the equation and considering any restrictions.

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