Answer :
To determine which of the provided options is equivalent to the function [tex]\( f(x) = 3(x + 10) \)[/tex], we need to carefully analyze each choice:
Given function: [tex]\( f(x) = 3(x + 10) \)[/tex]
Option A: [tex]\( y = 3(x + 10) \)[/tex]
To compare this to the given function, we set:
[tex]\[ f(x) = 3(x + 10) \][/tex]
and
[tex]\[ y = 3(x + 10) \][/tex]
Here, [tex]\( y \)[/tex] essentially represents [tex]\( f(x) \)[/tex]. This option matches the equation of [tex]\( f(x) \)[/tex], making it a valid representation.
Option B: [tex]\( y = 3f(x) - 10 \)[/tex]
We substitute [tex]\( f(x) \)[/tex] into this equation:
[tex]\[ y = 3[3(x + 10)] - 10 \][/tex]
[tex]\[ y = 3 \cdot 3(x + 10) - 10 \][/tex]
[tex]\[ y = 9(x + 10) - 10 \][/tex]
[tex]\[ y = 9x + 90 - 10 \][/tex]
[tex]\[ y = 9x + 80 \][/tex]
This result is different from the function [tex]\( f(x) = 3(x + 10) \)[/tex], so this option is incorrect.
Option C: [tex]\( y = 3x + 10 \)[/tex]
This equation is:
[tex]\[ y = 3x + 10 \][/tex]
This result does not match the given function [tex]\( f(x) = 3(x + 10) \)[/tex], as it would yield:
[tex]\( f(x) = 3x + 30 \)[/tex] if expanded. Hence, this option is incorrect.
Option D: [tex]\( y = 3f(x) - 30 \)[/tex]
Substitute [tex]\( f(x) \)[/tex] into this equation:
[tex]\[ y = 3[3(x + 10)] - 30 \][/tex]
[tex]\[ y = 3 \cdot 3(x + 10) - 30 \][/tex]
[tex]\[ y = 9(x + 10) - 30 \][/tex]
[tex]\[ y = 9x + 90 - 30 \][/tex]
[tex]\[ y = 9x + 60 \][/tex]
This result is different from the function [tex]\( f(x) = 3(x + 10) \)[/tex], so this option is incorrect.
Conclusion:
The correct answer is Option A: [tex]\( y = 3(x + 10) \)[/tex].
Thus, the function written in equation notation that is equivalent to the function [tex]\( f(x) = 3(x + 10) \)[/tex] is:
A. [tex]\( y = 3(x + 10) \)[/tex].
Given function: [tex]\( f(x) = 3(x + 10) \)[/tex]
Option A: [tex]\( y = 3(x + 10) \)[/tex]
To compare this to the given function, we set:
[tex]\[ f(x) = 3(x + 10) \][/tex]
and
[tex]\[ y = 3(x + 10) \][/tex]
Here, [tex]\( y \)[/tex] essentially represents [tex]\( f(x) \)[/tex]. This option matches the equation of [tex]\( f(x) \)[/tex], making it a valid representation.
Option B: [tex]\( y = 3f(x) - 10 \)[/tex]
We substitute [tex]\( f(x) \)[/tex] into this equation:
[tex]\[ y = 3[3(x + 10)] - 10 \][/tex]
[tex]\[ y = 3 \cdot 3(x + 10) - 10 \][/tex]
[tex]\[ y = 9(x + 10) - 10 \][/tex]
[tex]\[ y = 9x + 90 - 10 \][/tex]
[tex]\[ y = 9x + 80 \][/tex]
This result is different from the function [tex]\( f(x) = 3(x + 10) \)[/tex], so this option is incorrect.
Option C: [tex]\( y = 3x + 10 \)[/tex]
This equation is:
[tex]\[ y = 3x + 10 \][/tex]
This result does not match the given function [tex]\( f(x) = 3(x + 10) \)[/tex], as it would yield:
[tex]\( f(x) = 3x + 30 \)[/tex] if expanded. Hence, this option is incorrect.
Option D: [tex]\( y = 3f(x) - 30 \)[/tex]
Substitute [tex]\( f(x) \)[/tex] into this equation:
[tex]\[ y = 3[3(x + 10)] - 30 \][/tex]
[tex]\[ y = 3 \cdot 3(x + 10) - 30 \][/tex]
[tex]\[ y = 9(x + 10) - 30 \][/tex]
[tex]\[ y = 9x + 90 - 30 \][/tex]
[tex]\[ y = 9x + 60 \][/tex]
This result is different from the function [tex]\( f(x) = 3(x + 10) \)[/tex], so this option is incorrect.
Conclusion:
The correct answer is Option A: [tex]\( y = 3(x + 10) \)[/tex].
Thus, the function written in equation notation that is equivalent to the function [tex]\( f(x) = 3(x + 10) \)[/tex] is:
A. [tex]\( y = 3(x + 10) \)[/tex].
