How to tell end behavior of a function?
Understand the Problem
The question is asking for guidance on how to determine the end behavior of a mathematical function, which involves analyzing how the function behaves as the input values approach positive or negative infinity.
Answer
As $x \to +\infty$, $f(x) \to +\infty$; as $x \to -\infty$, $f(x) \to -\infty$.
Answer for screen readers
As $x \to +\infty$, $f(x) \to +\infty$; and as $x \to -\infty$, $f(x) \to -\infty$.
Steps to Solve
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Identify the function First, determine the mathematical function you are analyzing. For example, let's consider a polynomial function like $f(x) = 2x^3 - 4x + 1$.
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Determine the leading term Find the term in the function with the highest degree, as this term will primarily influence the end behavior. In our example, the leading term is $2x^3$.
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Analyze the leading coefficient and degree
- If the leading coefficient is positive and the degree is odd, the function will approach positive infinity as $x$ approaches positive infinity and negative infinity as $x$ approaches negative infinity.
- If the leading coefficient is negative and the degree is odd, the function will approach negative infinity as $x$ approaches positive infinity and positive infinity as $x$ approaches negative infinity.
- For an even degree, if the leading coefficient is positive, the function will approach positive infinity in both directions, while if it is negative, it will approach negative infinity in both directions.
Using our example: the leading term $2x^3$ has a positive coefficient and is of odd degree.
- Conclude the end behavior Based on the analysis of the leading term, conclude the end behavior of the function:
- As $x \to +\infty$, $f(x) \to +\infty$.
- As $x \to -\infty$, $f(x) \to -\infty$.
As $x \to +\infty$, $f(x) \to +\infty$; and as $x \to -\infty$, $f(x) \to -\infty$.
More Information
The end behavior of a function is important in understanding its overall trend, especially when graphing it or solving limits as $x$ approaches infinity.
Tips
- Misidentifying the leading term or its coefficient.
- Forgetting that odd/even degree affects signs of infinity differently.
- Analyzing the wrong function entirely.
