Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Explore in-depth answers to your questions from a knowledgeable community of experts across different fields. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
To find the resulting function after applying the given sequence of transformations to [tex]\( f(x) = x^5 \)[/tex], let's follow each transformation step-by-step:
1. Vertical Compression by [tex]\(\frac{1}{2}\)[/tex]:
When we compress a function vertically by a factor of [tex]\(\frac{1}{2}\)[/tex], we multiply the entire function by [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[ f(x) = x^5 \implies \frac{1}{2} \cdot x^5 \][/tex]
Therefore, after this transformation, our modified function is:
[tex]\[ g(x) = \frac{1}{2} x^5 \][/tex]
2. Horizontal Shift Left by 2 Units:
To shift the function horizontally to the left by 2 units, we replace [tex]\( x \)[/tex] with [tex]\( x + 2 \)[/tex] in the function:
[tex]\[ g(x) = \frac{1}{2}(x + 2)^5 \][/tex]
3. Vertical Shift Down by 1 Unit:
To shift the function vertically down by 1 unit, we subtract 1 from the function:
[tex]\[ g(x) = \frac{1}{2}(x + 2)^5 - 1 \][/tex]
Combining all these transformations, we get the resulting function after applying the sequence of transformations to [tex]\( f(x) = x^5 \)[/tex]:
[tex]\[ g(x) = \frac{1}{2}(x + 2)^5 - 1 \][/tex]
Thus, the correct answer is:
B. [tex]\( g(x) = \frac{1}{2}(x + 2)^5 - 1 \)[/tex]
1. Vertical Compression by [tex]\(\frac{1}{2}\)[/tex]:
When we compress a function vertically by a factor of [tex]\(\frac{1}{2}\)[/tex], we multiply the entire function by [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[ f(x) = x^5 \implies \frac{1}{2} \cdot x^5 \][/tex]
Therefore, after this transformation, our modified function is:
[tex]\[ g(x) = \frac{1}{2} x^5 \][/tex]
2. Horizontal Shift Left by 2 Units:
To shift the function horizontally to the left by 2 units, we replace [tex]\( x \)[/tex] with [tex]\( x + 2 \)[/tex] in the function:
[tex]\[ g(x) = \frac{1}{2}(x + 2)^5 \][/tex]
3. Vertical Shift Down by 1 Unit:
To shift the function vertically down by 1 unit, we subtract 1 from the function:
[tex]\[ g(x) = \frac{1}{2}(x + 2)^5 - 1 \][/tex]
Combining all these transformations, we get the resulting function after applying the sequence of transformations to [tex]\( f(x) = x^5 \)[/tex]:
[tex]\[ g(x) = \frac{1}{2}(x + 2)^5 - 1 \][/tex]
Thus, the correct answer is:
B. [tex]\( g(x) = \frac{1}{2}(x + 2)^5 - 1 \)[/tex]
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.