Introduction
In mathematics, a function is a relation between a set of inputs and a set of possible outputs with the property that each input is related to exactly one output. The range of a function is the set of all possible output values that the function can produce. In this article, we will discuss how to identify the range of a function shown in a graph.
Understanding the Graph of a Function
A graph of a function is a visual representation of the relationship between the input and output values of the function. The horizontal axis of the graph represents the input values, while the vertical axis represents the output values. The graph of a function can be used to determine the domain and range of the function.
Example:
Consider the function f(x) = x^2. The graph of this function is shown below:
The horizontal axis represents the input values, which in this case are the values of x. The vertical axis represents the output values, which in this case are the values of f(x). The graph shows that for every value of x, there is a corresponding value of f(x). The range of this function is all non-negative real numbers, since the function can produce any non-negative value.
Identifying the Range of a Function from a Graph
To identify the range of a function from a graph, we need to look at the vertical axis of the graph and determine the set of all possible output values. The range of the function is the set of all output values that the function can produce.
Example:
Consider the function g(x) = sin(x). The graph of this function is shown below:
The graph shows that the output values of the function g(x) range from -1 to 1. Therefore, the range of the function g(x) is [-1, 1].
Using Calculus to Find the Range of a Function
In some cases, it may be difficult to determine the range of a function from a graph. In these cases, we can use calculus to find the range of the function.
Example:
Consider the function h(x) = x^3 – 3x^2 + 2x. To find the range of this function, we can use calculus. First, we find the derivative of the function:
h'(x) = 3x^2 – 6x + 2
Next, we find the critical points of the function by setting the derivative equal to zero:
3x^2 – 6x + 2 = 0
Solving for x, we get:
x = 1 ± sqrt(2)/3
These are the critical points of the function. We can now use the second derivative test to determine whether these points are local maxima or minima. The second derivative of the function is:
h”(x) = 6x – 6
Plugging in the critical points, we get:
h”(1 + sqrt(2)/3) = 2sqrt(2) – 6 < 0
h”(1 – sqrt(2)/3) = -2sqrt(2) – 6 < 0
Since the second derivative is negative at both critical points, they are both local maxima. Therefore, the range of the function is:
range(h) = (-∞, h(1 + sqrt(2)/3)] ∪ [h(1 – sqrt(2)/3), ∞)
Conclusion
In conclusion, the range of a function is the set of all possible output values that the function can produce. To identify the range of a function from a graph, we need to look at the vertical axis of the graph and determine the set of all possible output values. In some cases, we may need to use calculus to find the range of a function. By understanding how to identify the range of a function, we can better understand the behavior of the function and its relationship between input and output values.
