Finding the range of a function in Class 11 involves determining all possible output values (y-values) the function can produce. Here's how you can approach this:
Steps to Determine the Range of a Function
The following steps outline how to find the range of a function, based on the reference provided:
- Draw the graph of the given function. Visualizing the function through its graph is often the most intuitive way to understand its behavior.
- Note down the minimum and maximum value of the variable y on the y-axis over which the graph is spread. Identify the lowest and highest y-values that the graph attains.
- The range of the function will be [minimum y value, maximum y value]. Express the range as an interval between the minimum and maximum y-values. This interval includes all possible output values of the function.
Example
Let's consider a simple example to illustrate this process.
Suppose we want to find the range of the function f(x) = x2
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Graph: The graph of f(x) = x2 is a parabola opening upwards, with its vertex at the origin (0,0).
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Minimum and Maximum y-value: The minimum y-value is 0 (at the vertex). As x moves away from 0 in either direction, x2 becomes increasingly large, without any upper bound. Thus, there is no maximum y-value.
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Range: The range of the function is [0, ∞). This means the function's output can be any non-negative real number.
Additional Tips
- Consider the Domain: The domain of the function (possible input values) influences the range. Restrictions on the domain will affect the possible output values.
- Analyze Function Behavior: Understand how the function behaves as x approaches positive and negative infinity, and around any points of discontinuity.
- Use Algebraic Manipulation: Sometimes, rearranging the function can help reveal its range. For example, completing the square in a quadratic function makes it easy to see the minimum or maximum value.
By following these steps, you can effectively determine the range of a function in Class 11.
