Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Discover solutions to your questions from experienced professionals across multiple fields on our comprehensive Q&A platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
To determine the end behavior of the graph of [tex]\( f(x) = x^3 (x+3) (-5x+1) \)[/tex] using limits, we'll analyze the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex] and [tex]\( +\infty \)[/tex].
1. Examining [tex]\( f(x) \)[/tex] as [tex]\( x \to -\infty \)[/tex]:
- At a large negative [tex]\( x \)[/tex], [tex]\( x \)[/tex] is a large negative number, [tex]\( x + 3 \)[/tex] is also large and negative, and [tex]\( -5x + 1 \)[/tex] becomes positive since [tex]\( -5 \times \text{negative\_large\_number} \)[/tex] will be positive.
- Therefore, multiplying these together [tex]\(( x^3 (x+3) (-5x+1) )\)[/tex], the product of two negative numbers (which is positive) and one positive number is positive. Since the term with the highest degree, [tex]\( x^5 \)[/tex], has a negative coefficient, the overall function will be highly negative.
Hence, as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
2. Examining [tex]\( f(x) \)[/tex] as [tex]\( x \to +\infty \)[/tex]:
- At a large positive [tex]\( x \)[/tex], [tex]\( x \)[/tex] is a large positive number, [tex]\( x + 3 \)[/tex] is also a large positive number, and [tex]\( -5x + 1 \)[/tex] becomes negative since [tex]\( -5 \times \text{positive\_large\_number} \)[/tex] will be a large negative.
- Therefore, multiplying these together [tex]\(( x^3 (x+3) (-5x+1))\)[/tex], the product of three positive numbers (two in the significant domain of negative) and one negative number is negative.
- Similarly, since the term with the highest degree, [tex]\( x^5 \)[/tex], has a negative coefficient, the function will be highly negative.
Hence, as [tex]\( x \to +\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
To summarize:
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to +\infty \)[/tex].
- As [tex]\( x \to +\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
So, the function exhibits the following end behavior:
[tex]\[ \text{As } x \rightarrow -\infty, f(x) \rightarrow -\infty \][/tex]
[tex]\[ \text{As } x \rightarrow +\infty, f(x) \rightarrow -\infty \][/tex]
[tex]\[ \text{As } x \rightarrow -\infty, f(x) \rightarrow +\infty \][/tex]
[tex]\[ \text{As } x \rightarrow +\infty, f(x) \rightarrow \infty \][/tex]
[tex]\[ \text{As } x \rightarrow -\infty, f(x) \rightarrow +\infty \][/tex]
[tex]\[ \text{As } x \rightarrow +\infty, f(x) \rightarrow \infty \][/tex]
1. Examining [tex]\( f(x) \)[/tex] as [tex]\( x \to -\infty \)[/tex]:
- At a large negative [tex]\( x \)[/tex], [tex]\( x \)[/tex] is a large negative number, [tex]\( x + 3 \)[/tex] is also large and negative, and [tex]\( -5x + 1 \)[/tex] becomes positive since [tex]\( -5 \times \text{negative\_large\_number} \)[/tex] will be positive.
- Therefore, multiplying these together [tex]\(( x^3 (x+3) (-5x+1) )\)[/tex], the product of two negative numbers (which is positive) and one positive number is positive. Since the term with the highest degree, [tex]\( x^5 \)[/tex], has a negative coefficient, the overall function will be highly negative.
Hence, as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
2. Examining [tex]\( f(x) \)[/tex] as [tex]\( x \to +\infty \)[/tex]:
- At a large positive [tex]\( x \)[/tex], [tex]\( x \)[/tex] is a large positive number, [tex]\( x + 3 \)[/tex] is also a large positive number, and [tex]\( -5x + 1 \)[/tex] becomes negative since [tex]\( -5 \times \text{positive\_large\_number} \)[/tex] will be a large negative.
- Therefore, multiplying these together [tex]\(( x^3 (x+3) (-5x+1))\)[/tex], the product of three positive numbers (two in the significant domain of negative) and one negative number is negative.
- Similarly, since the term with the highest degree, [tex]\( x^5 \)[/tex], has a negative coefficient, the function will be highly negative.
Hence, as [tex]\( x \to +\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
To summarize:
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to +\infty \)[/tex].
- As [tex]\( x \to +\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
So, the function exhibits the following end behavior:
[tex]\[ \text{As } x \rightarrow -\infty, f(x) \rightarrow -\infty \][/tex]
[tex]\[ \text{As } x \rightarrow +\infty, f(x) \rightarrow -\infty \][/tex]
[tex]\[ \text{As } x \rightarrow -\infty, f(x) \rightarrow +\infty \][/tex]
[tex]\[ \text{As } x \rightarrow +\infty, f(x) \rightarrow \infty \][/tex]
[tex]\[ \text{As } x \rightarrow -\infty, f(x) \rightarrow +\infty \][/tex]
[tex]\[ \text{As } x \rightarrow +\infty, f(x) \rightarrow \infty \][/tex]
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
