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To determine which function has a range of [tex]\( y < 3 \)[/tex], let's analyze each function one by one and discuss their ranges.
1. Function: [tex]\( y = 3(2)^x \)[/tex]
This function is an exponential function where the base is 2 and it is scaled by a factor of 3. Since [tex]\( 2^x \)[/tex] is always positive for any real number [tex]\( x \)[/tex], multiplying it by 3 will give:
[tex]\[ y = 3 \times (positive \ value) = positive \ value \][/tex]
Therefore, the range of this function is [tex]\( y > 0 \)[/tex]. Clearly, this does not satisfy [tex]\( y < 3 \)[/tex].
2. Function: [tex]\( y = 2(3)^x \)[/tex]
Similar to the first function, this is also an exponential function but with the base 3, scaled by a factor of 2. The expression [tex]\( 3^x \)[/tex] is always positive for any real number [tex]\( x \)[/tex], so:
[tex]\[ y = 2 \times (positive \ value) = positive \ value \][/tex]
The range of this function is [tex]\( y > 0 \)[/tex]. Hence, this does not meet the condition [tex]\( y < 3 \)[/tex].
3. Function: [tex]\( y = -(2)^x + 3 \)[/tex]
This function is also an exponential function with the base 2, but it has a negative coefficient and an addition of 3. Since [tex]\( 2^x \)[/tex] is always positive for any real number [tex]\( x \)[/tex], let's consider [tex]\( z = 2^x \)[/tex] where [tex]\( z > 0 \)[/tex]:
[tex]\[ y = -z + 3 \][/tex]
Here, as [tex]\( z \)[/tex] increases, [tex]\( -z \)[/tex] becomes more negative, keeping [tex]\( y \)[/tex] less than 3. The maximum value of [tex]\( y \)[/tex] occurs when [tex]\( z \)[/tex] is at its lowest, i.e., closest to 0 (but never actually reaching 0 because [tex]\( 2^x \)[/tex] is never zero):
[tex]\[ y = -0 + 3 = 3 \][/tex]
As [tex]\( z \)[/tex] increases, [tex]\( y \)[/tex] approaches 3 but is always less than 3. Therefore, the range of this function is [tex]\( y < 3 \)[/tex].
4. Function: [tex]\( y = (2)^x - 3 \)[/tex]
This is an exponential function with base 2, shifted downward by 3 units. As [tex]\( 2^x \)[/tex] is always positive, let [tex]\( z = 2^x \)[/tex]:
[tex]\[ y = z - 3 \][/tex]
Here, as [tex]\( z \)[/tex] increases, [tex]\( y \)[/tex] will also increase. The minimum value of [tex]\( y \)[/tex] occurs when [tex]\( z \)[/tex] is at its lowest, i.e., closest to 0:
[tex]\[ y = 0 - 3 = -3 \][/tex]
As [tex]\( z \)[/tex] grows, so does [tex]\( y \)[/tex]. Consequently, the range of this function is [tex]\( y \ge -3 \)[/tex]. There is no upper bound, hence this function does not have a finite range that stays under 3.
After evaluating the ranges of all the functions, it's clear that the function which has a range of [tex]\( y < 3 \)[/tex] is:
[tex]\[ y = -(2)^x + 3 \][/tex]
1. Function: [tex]\( y = 3(2)^x \)[/tex]
This function is an exponential function where the base is 2 and it is scaled by a factor of 3. Since [tex]\( 2^x \)[/tex] is always positive for any real number [tex]\( x \)[/tex], multiplying it by 3 will give:
[tex]\[ y = 3 \times (positive \ value) = positive \ value \][/tex]
Therefore, the range of this function is [tex]\( y > 0 \)[/tex]. Clearly, this does not satisfy [tex]\( y < 3 \)[/tex].
2. Function: [tex]\( y = 2(3)^x \)[/tex]
Similar to the first function, this is also an exponential function but with the base 3, scaled by a factor of 2. The expression [tex]\( 3^x \)[/tex] is always positive for any real number [tex]\( x \)[/tex], so:
[tex]\[ y = 2 \times (positive \ value) = positive \ value \][/tex]
The range of this function is [tex]\( y > 0 \)[/tex]. Hence, this does not meet the condition [tex]\( y < 3 \)[/tex].
3. Function: [tex]\( y = -(2)^x + 3 \)[/tex]
This function is also an exponential function with the base 2, but it has a negative coefficient and an addition of 3. Since [tex]\( 2^x \)[/tex] is always positive for any real number [tex]\( x \)[/tex], let's consider [tex]\( z = 2^x \)[/tex] where [tex]\( z > 0 \)[/tex]:
[tex]\[ y = -z + 3 \][/tex]
Here, as [tex]\( z \)[/tex] increases, [tex]\( -z \)[/tex] becomes more negative, keeping [tex]\( y \)[/tex] less than 3. The maximum value of [tex]\( y \)[/tex] occurs when [tex]\( z \)[/tex] is at its lowest, i.e., closest to 0 (but never actually reaching 0 because [tex]\( 2^x \)[/tex] is never zero):
[tex]\[ y = -0 + 3 = 3 \][/tex]
As [tex]\( z \)[/tex] increases, [tex]\( y \)[/tex] approaches 3 but is always less than 3. Therefore, the range of this function is [tex]\( y < 3 \)[/tex].
4. Function: [tex]\( y = (2)^x - 3 \)[/tex]
This is an exponential function with base 2, shifted downward by 3 units. As [tex]\( 2^x \)[/tex] is always positive, let [tex]\( z = 2^x \)[/tex]:
[tex]\[ y = z - 3 \][/tex]
Here, as [tex]\( z \)[/tex] increases, [tex]\( y \)[/tex] will also increase. The minimum value of [tex]\( y \)[/tex] occurs when [tex]\( z \)[/tex] is at its lowest, i.e., closest to 0:
[tex]\[ y = 0 - 3 = -3 \][/tex]
As [tex]\( z \)[/tex] grows, so does [tex]\( y \)[/tex]. Consequently, the range of this function is [tex]\( y \ge -3 \)[/tex]. There is no upper bound, hence this function does not have a finite range that stays under 3.
After evaluating the ranges of all the functions, it's clear that the function which has a range of [tex]\( y < 3 \)[/tex] is:
[tex]\[ y = -(2)^x + 3 \][/tex]
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