Question
Which statement correctly describes the function?
f left-parenthesis x right-parenthesis equals start fraction x plus 4 over x squared plus 7 x plus 10 end fraction
(1 point)
Responses
It is continuous.
It is continuous.
It is discontinuous because there is a value a for which f left-parenthesis a right-parenthesis is not defined.
It is discontinuous because there is a value a for which Image with alt text: f left-parenthesis a right-parenthesis is not defined.
It is discontinuous because there is a value a such that limit underscript x right arrow a endscript f left parenthesis x right parenthesis not equal to f left parenthesis x right parenthesis.
It is discontinuous because there is a value a such that Image with alt text: limit underscript x right arrow a endscript f left parenthesis x right parenthesis not equal to f left parenthesis x right parenthesis .
It is discontinuous because there is a value a such that limit underscript x right arrow a endscript f left parenthesis x right parenthesis does not exist.
f left-parenthesis x right-parenthesis equals start fraction x plus 4 over x squared plus 7 x plus 10 end fraction
(1 point)
Responses
It is continuous.
It is continuous.
It is discontinuous because there is a value a for which f left-parenthesis a right-parenthesis is not defined.
It is discontinuous because there is a value a for which Image with alt text: f left-parenthesis a right-parenthesis is not defined.
It is discontinuous because there is a value a such that limit underscript x right arrow a endscript f left parenthesis x right parenthesis not equal to f left parenthesis x right parenthesis.
It is discontinuous because there is a value a such that Image with alt text: limit underscript x right arrow a endscript f left parenthesis x right parenthesis not equal to f left parenthesis x right parenthesis .
It is discontinuous because there is a value a such that limit underscript x right arrow a endscript f left parenthesis x right parenthesis does not exist.
Answers
Answered by
GPT-4o mini
To analyze the function \( f(x) = \frac{x + 4}{x^2 + 7x + 10} \), we need to determine its continuity.
First, we look for points where the function is defined. The function will be undefined when the denominator is equal to zero:
\[
x^2 + 7x + 10 = 0
\]
Factoring the quadratic:
\[
(x + 2)(x + 5) = 0
\]
This gives us the points \( x = -2 \) and \( x = -5 \). At these points, the function \( f(x) \) is undefined.
Now, let's evaluate the continuity:
1. A function is continuous at a point \( a \) if three conditions hold:
- \( f(a) \) is defined.
- The limit of \( f(x) \) as \( x \) approaches \( a \) exists.
- The limit equals \( f(a) \).
2. Since \( f(x) \) is undefined at \( x = -2 \) and \( x = -5 \), the function cannot be continuous at these points. Therefore, it cannot be continuous overall because there are points of discontinuity.
Given the analysis, the best option among those provided is:
**It is discontinuous because there is a value \( a \) for which \( f(a) \) is not defined.**
First, we look for points where the function is defined. The function will be undefined when the denominator is equal to zero:
\[
x^2 + 7x + 10 = 0
\]
Factoring the quadratic:
\[
(x + 2)(x + 5) = 0
\]
This gives us the points \( x = -2 \) and \( x = -5 \). At these points, the function \( f(x) \) is undefined.
Now, let's evaluate the continuity:
1. A function is continuous at a point \( a \) if three conditions hold:
- \( f(a) \) is defined.
- The limit of \( f(x) \) as \( x \) approaches \( a \) exists.
- The limit equals \( f(a) \).
2. Since \( f(x) \) is undefined at \( x = -2 \) and \( x = -5 \), the function cannot be continuous at these points. Therefore, it cannot be continuous overall because there are points of discontinuity.
Given the analysis, the best option among those provided is:
**It is discontinuous because there is a value \( a \) for which \( f(a) \) is not defined.**
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