A function is increasing when…
A function is considered increasing when the value of the function rises as the input variable increases
A function is considered increasing when the value of the function rises as the input variable increases. In other words, as the x-values in the domain of the function increase, the corresponding y-values in the range also increase.
To determine if a function is increasing, you can use the following criteria:
1. Slope: The slope of the function’s graph is positive. This means that the tangent line to the graph, at any point, has a positive slope. If you find that the slope of the function is always positive, then the function is increasing.
2. First Derivative: The first derivative of the function is positive. The first derivative represents the rate of change of the function. If the first derivative is positive for all values in the domain, then the function is increasing.
3. Difference in Y-values: The y-values of the function increase as the x-values increase. For different x-values, if the corresponding y-values are always larger, then the function is increasing.
Keep in mind that the function can be increasing either over a specific interval or across its entire domain. In some cases, the function might be increasing on one interval and decreasing on another.
It’s important to analyze the function using all available methods to confirm that it is indeed increasing. This ensures accuracy and a thorough understanding of the behavior of the function.
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