Answer :
To determine which function has a range of [tex]\( y < 3 \)[/tex], let's evaluate each of the given functions:
1. Function: [tex]\( y = 3(2)^x \)[/tex]
- For any real number [tex]\( x \)[/tex], the expression [tex]\( 2^x \)[/tex] is always positive, giving values greater than 1 when [tex]\( x \)[/tex] is positive and values between 0 and 1 when [tex]\( x \)[/tex] is negative.
- Therefore, [tex]\( 3(2)^x \)[/tex] is always positive and greater than 0.
- The range is [tex]\( y > 0 \)[/tex].
2. Function: [tex]\( y = 2(3)^x \)[/tex]
- For any real number [tex]\( x \)[/tex], the expression [tex]\( 3^x \)[/tex] is also always positive, leading to values greater than 3 when [tex]\( x \)[/tex] is positive and values between 0 and 1 when [tex]\( x \)[/tex] is negative.
- Thus, [tex]\( 2(3)^x \)[/tex] is always positive and greater than 0.
- The range is [tex]\( y > 0 \)[/tex].
3. Function: [tex]\( y = -(2)^x + 3 \)[/tex]
- For any real number [tex]\( x \)[/tex], the term [tex]\( 2^x \)[/tex] is positive, making [tex]\( -(2)^x \)[/tex] negative.
- When you add 3 to a negative value, the resulting value is less than 3.
- Thus, the function [tex]\( y = -(2)^x + 3 \)[/tex] yields values [tex]\( y < 3 \)[/tex].
4. Function: [tex]\( y = (2)^x - 3 \)[/tex]
- For any real number [tex]\( x \)[/tex], the term [tex]\( 2^x \)[/tex] is positive.
- Subtracting 3 from a positive value can yield positive values starting from [tex]\( -3 \)[/tex] upward.
- The range is [tex]\( y > -3 \)[/tex].
Out of the evaluated functions, the third function [tex]\( y = -(2)^x + 3 \)[/tex] is the one that gives a range of [tex]\( y < 3 \)[/tex].
Hence, the function that has a range of [tex]\( y < 3 \)[/tex] is:
[tex]\[ y = -(2)^x + 3 \][/tex]
1. Function: [tex]\( y = 3(2)^x \)[/tex]
- For any real number [tex]\( x \)[/tex], the expression [tex]\( 2^x \)[/tex] is always positive, giving values greater than 1 when [tex]\( x \)[/tex] is positive and values between 0 and 1 when [tex]\( x \)[/tex] is negative.
- Therefore, [tex]\( 3(2)^x \)[/tex] is always positive and greater than 0.
- The range is [tex]\( y > 0 \)[/tex].
2. Function: [tex]\( y = 2(3)^x \)[/tex]
- For any real number [tex]\( x \)[/tex], the expression [tex]\( 3^x \)[/tex] is also always positive, leading to values greater than 3 when [tex]\( x \)[/tex] is positive and values between 0 and 1 when [tex]\( x \)[/tex] is negative.
- Thus, [tex]\( 2(3)^x \)[/tex] is always positive and greater than 0.
- The range is [tex]\( y > 0 \)[/tex].
3. Function: [tex]\( y = -(2)^x + 3 \)[/tex]
- For any real number [tex]\( x \)[/tex], the term [tex]\( 2^x \)[/tex] is positive, making [tex]\( -(2)^x \)[/tex] negative.
- When you add 3 to a negative value, the resulting value is less than 3.
- Thus, the function [tex]\( y = -(2)^x + 3 \)[/tex] yields values [tex]\( y < 3 \)[/tex].
4. Function: [tex]\( y = (2)^x - 3 \)[/tex]
- For any real number [tex]\( x \)[/tex], the term [tex]\( 2^x \)[/tex] is positive.
- Subtracting 3 from a positive value can yield positive values starting from [tex]\( -3 \)[/tex] upward.
- The range is [tex]\( y > -3 \)[/tex].
Out of the evaluated functions, the third function [tex]\( y = -(2)^x + 3 \)[/tex] is the one that gives a range of [tex]\( y < 3 \)[/tex].
Hence, the function that has a range of [tex]\( y < 3 \)[/tex] is:
[tex]\[ y = -(2)^x + 3 \][/tex]
