Question
Describe the end behavior of the function below.
(2 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
As x approaches −∞, f(x) approaches
Response area.
As x approaches ∞, f(x) approaches
Response area.
(2 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
As x approaches −∞, f(x) approaches
Response area.
As x approaches ∞, f(x) approaches
Response area.
Answers
Answered by
GPT-4o mini
To describe the end behavior of a function, we need to look at its leading term or general behavior of the function. While you haven't provided the specific function, I can guide you on how to analyze it based on its degree and leading coefficients.
1. **For polynomial functions**:
- If the leading coefficient is **positive** and the degree is **even**, as \( x \) approaches \( -\infty \), \( f(x) \) approaches \( +\infty \); as \( x \) approaches \( +\infty \), \( f(x) \) approaches \( +\infty \).
- If the leading coefficient is **negative** and the degree is **even**, as \( x \) approaches \( -\infty \), \( f(x) \) approaches \( -\infty \); as \( x \) approaches \( +\infty \), \( f(x) \) approaches \( -\infty \).
- If the leading coefficient is **positive** and the degree is **odd**, as \( x \) approaches \( -\infty \), \( f(x) \) approaches \( -\infty \); as \( x \) approaches \( +\infty \), \( f(x) \) approaches \( +\infty \).
- If the leading coefficient is **negative** and the degree is **odd**, as \( x \) approaches \( -\infty \), \( f(x) \) approaches \( +\infty \); as \( x \) approaches \( +\infty \), \( f(x) \) approaches \( -\infty \).
2. **For rational functions**:
- You can analyze end behavior by finding the degrees of the numerator and denominator.
3. **For exponential functions** (like \( f(x) = a^x \)):
- If \( a > 1 \), as \( x \) approaches \( -\infty \), \( f(x) \) approaches 0; as \( x \) approaches \( +\infty \), \( f(x) \) approaches \( +\infty \).
- If \( 0 < a < 1 \), as \( x \) approaches \( -\infty \), \( f(x) \) approaches \( +\infty \); as \( x \) approaches \( +\infty \), \( f(x) \) approaches 0.
If you have a particular function in mind, please share it, and I can give you explicit end behavior responses for that function. Otherwise, you can fill in your responses as follows based on the function you are analyzing:
- As \( x \) approaches \( -\infty \), \( f(x) \) approaches [insert value based on analysis].
- As \( x \) approaches \( +\infty \), \( f(x) \) approaches [insert value based on analysis].
1. **For polynomial functions**:
- If the leading coefficient is **positive** and the degree is **even**, as \( x \) approaches \( -\infty \), \( f(x) \) approaches \( +\infty \); as \( x \) approaches \( +\infty \), \( f(x) \) approaches \( +\infty \).
- If the leading coefficient is **negative** and the degree is **even**, as \( x \) approaches \( -\infty \), \( f(x) \) approaches \( -\infty \); as \( x \) approaches \( +\infty \), \( f(x) \) approaches \( -\infty \).
- If the leading coefficient is **positive** and the degree is **odd**, as \( x \) approaches \( -\infty \), \( f(x) \) approaches \( -\infty \); as \( x \) approaches \( +\infty \), \( f(x) \) approaches \( +\infty \).
- If the leading coefficient is **negative** and the degree is **odd**, as \( x \) approaches \( -\infty \), \( f(x) \) approaches \( +\infty \); as \( x \) approaches \( +\infty \), \( f(x) \) approaches \( -\infty \).
2. **For rational functions**:
- You can analyze end behavior by finding the degrees of the numerator and denominator.
3. **For exponential functions** (like \( f(x) = a^x \)):
- If \( a > 1 \), as \( x \) approaches \( -\infty \), \( f(x) \) approaches 0; as \( x \) approaches \( +\infty \), \( f(x) \) approaches \( +\infty \).
- If \( 0 < a < 1 \), as \( x \) approaches \( -\infty \), \( f(x) \) approaches \( +\infty \); as \( x \) approaches \( +\infty \), \( f(x) \) approaches 0.
If you have a particular function in mind, please share it, and I can give you explicit end behavior responses for that function. Otherwise, you can fill in your responses as follows based on the function you are analyzing:
- As \( x \) approaches \( -\infty \), \( f(x) \) approaches [insert value based on analysis].
- As \( x \) approaches \( +\infty \), \( f(x) \) approaches [insert value based on analysis].