Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Get immediate and reliable answers to your questions from a community of experienced experts on our platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
Let's analyze how modifying the function [tex]\( f(x) = a b^x \)[/tex] by increasing the value of [tex]\( a \)[/tex] by 2 affects its domain and range.
### Domain Analysis
The original function [tex]\( f(x) = a b^x \)[/tex] is an exponential function. For exponential functions, the independent variable [tex]\( x \)[/tex] can take any real number value. This means the domain of the function [tex]\( f(x) \)[/tex] is all real numbers, denoted as [tex]\( (-\infty, \infty) \)[/tex].
When we increase the value of [tex]\( a \)[/tex] by 2, the new function becomes [tex]\( f_{new}(x) = (a+2) b^x \)[/tex].
- Since the operation modifies only the coefficient [tex]\( a \)[/tex] and does not affect the exponent or the base [tex]\( b \)[/tex], the domain of the new function remains the same. That is, the new domain is still all real numbers, [tex]\( (-\infty, \infty) \)[/tex].
Thus, the statement "The domain stays the same" is True.
- There is no change to the domain that would restrict [tex]\( x \)[/tex] to values greater than 2 or greater than or equal to 2, so the statements "The domain becomes [tex]\( x > 2 \)[/tex]" and "The domain becomes [tex]\( x \geq 2 \)[/tex]" are both False.
### Range Analysis
Now let's consider the range of the function.
For the original function [tex]\( f(x) = a b^x \)[/tex]:
- If [tex]\( a > 0 \)[/tex] and [tex]\( b > 1 \)[/tex], the range of [tex]\( f(x) \)[/tex] is [tex]\( (0, \infty) \)[/tex] because [tex]\( b^x \)[/tex] is always positive and so is [tex]\( a \)[/tex].
- If [tex]\( a < 0 \)[/tex], the range would be [tex]\( (-\infty, 0) \)[/tex].
After increasing [tex]\( a \)[/tex] by 2, the new function is [tex]\( f_{new}(x) = (a+2) b^x \)[/tex].
- If the original [tex]\(a > 0\)[/tex]: The range of the original function was [tex]\( (0, \infty) \)[/tex]. With [tex]\(a+2\)[/tex], the range of the new function is now [tex]\( (2, \infty) \)[/tex].
- If the original [tex]\(a < 0\)[/tex]: The range of the original function was [tex]\( (-\infty, 0) \)[/tex]. With [tex]\(a+2\)[/tex], the range of the new function could potentially still be negative or include non-negative values, depending on [tex]\(a\)[/tex].
However, in a common scenario where [tex]\(a > 0\)[/tex], the range of the new function shifts upwards by 2 units.
Given the common case:
- The statement "The range stays the same" is False because the range shifts upwards.
- The statement "The range becomes [tex]\( y>2 \)[/tex]" is True because the entire range moves up by 2 units.
- The statement "The range becomes [tex]\( y \geq 2 \)[/tex]" is False because the range [tex]\( y > 2 \)[/tex] excludes 2 itself as [tex]\( f(x) \)[/tex] will never equal exactly 2.
In summary:
- The domain stays the same: True.
- The range stays the same: False.
- The range becomes [tex]\( y > 2 \)[/tex]: True.
- The domain becomes [tex]\( x > 2 \)[/tex]: False.
- The range becomes [tex]\( y \geq 2 \)[/tex]: False.
- The domain becomes [tex]\( x \geq 2 \)[/tex]: False.
Explicitly, the correct selections are:
- The range becomes [tex]\( y > 2 \)[/tex].
- The domain stays the same.
### Domain Analysis
The original function [tex]\( f(x) = a b^x \)[/tex] is an exponential function. For exponential functions, the independent variable [tex]\( x \)[/tex] can take any real number value. This means the domain of the function [tex]\( f(x) \)[/tex] is all real numbers, denoted as [tex]\( (-\infty, \infty) \)[/tex].
When we increase the value of [tex]\( a \)[/tex] by 2, the new function becomes [tex]\( f_{new}(x) = (a+2) b^x \)[/tex].
- Since the operation modifies only the coefficient [tex]\( a \)[/tex] and does not affect the exponent or the base [tex]\( b \)[/tex], the domain of the new function remains the same. That is, the new domain is still all real numbers, [tex]\( (-\infty, \infty) \)[/tex].
Thus, the statement "The domain stays the same" is True.
- There is no change to the domain that would restrict [tex]\( x \)[/tex] to values greater than 2 or greater than or equal to 2, so the statements "The domain becomes [tex]\( x > 2 \)[/tex]" and "The domain becomes [tex]\( x \geq 2 \)[/tex]" are both False.
### Range Analysis
Now let's consider the range of the function.
For the original function [tex]\( f(x) = a b^x \)[/tex]:
- If [tex]\( a > 0 \)[/tex] and [tex]\( b > 1 \)[/tex], the range of [tex]\( f(x) \)[/tex] is [tex]\( (0, \infty) \)[/tex] because [tex]\( b^x \)[/tex] is always positive and so is [tex]\( a \)[/tex].
- If [tex]\( a < 0 \)[/tex], the range would be [tex]\( (-\infty, 0) \)[/tex].
After increasing [tex]\( a \)[/tex] by 2, the new function is [tex]\( f_{new}(x) = (a+2) b^x \)[/tex].
- If the original [tex]\(a > 0\)[/tex]: The range of the original function was [tex]\( (0, \infty) \)[/tex]. With [tex]\(a+2\)[/tex], the range of the new function is now [tex]\( (2, \infty) \)[/tex].
- If the original [tex]\(a < 0\)[/tex]: The range of the original function was [tex]\( (-\infty, 0) \)[/tex]. With [tex]\(a+2\)[/tex], the range of the new function could potentially still be negative or include non-negative values, depending on [tex]\(a\)[/tex].
However, in a common scenario where [tex]\(a > 0\)[/tex], the range of the new function shifts upwards by 2 units.
Given the common case:
- The statement "The range stays the same" is False because the range shifts upwards.
- The statement "The range becomes [tex]\( y>2 \)[/tex]" is True because the entire range moves up by 2 units.
- The statement "The range becomes [tex]\( y \geq 2 \)[/tex]" is False because the range [tex]\( y > 2 \)[/tex] excludes 2 itself as [tex]\( f(x) \)[/tex] will never equal exactly 2.
In summary:
- The domain stays the same: True.
- The range stays the same: False.
- The range becomes [tex]\( y > 2 \)[/tex]: True.
- The domain becomes [tex]\( x > 2 \)[/tex]: False.
- The range becomes [tex]\( y \geq 2 \)[/tex]: False.
- The domain becomes [tex]\( x \geq 2 \)[/tex]: False.
Explicitly, the correct selections are:
- The range becomes [tex]\( y > 2 \)[/tex].
- The domain stays the same.
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.