Answer :
To determine which function has a range of \( y < 3 \), we need to analyze the ranges of each function given:
1. \( y = 3 \cdot (2)^x \)
- This function involves an exponential term with a positive base.
- As \( x \) increases, \( 2^x \) increases, and hence \( 3 \cdot (2)^x \) also increases.
- Exponential functions like \( 2^x \) are always positive and increasing for all real \( x \).
- Therefore, the range of this function is \( y > 0 \).
2. \( y = 2 \cdot (3)^x \)
- Similarly, this is an exponential function with a positive base.
- As \( x \) increases, \( 3^x \) increases, and \( 2 \cdot (3)^x \) also increases.
- Since \( 3^x \) is always positive and increasing, the range of this function is also \( y > 0 \).
3. \( y = -(2)^x + 3 \)
- Here, the function involves a negative exponential term added to 3.
- As \( x \) increases, \( 2^x \) increases, but since it is multiplied by \(-1\), the value \(-(2)^x\) becomes more negative.
- Therefore, the function can take values less than 3 because as \( x \) increases, the negative term will make \( y \) decrease.
- This implies the range of this function can include values less than 3. Specifically, since the maximum value \( y \) can achieve is 3 (when \( x = 0 \)), and it decreases from there, the range of this function is \( y < 3 \).
4. \( y = (2)^x - 3 \)
- This function involves a positive exponential term subtracted by 3.
- As \( x \) increases, \( 2^x \) increases, and hence \( (2)^x - 3 \) also increases.
- The minimum value this function can take is when \( x = 0 \), which gives \( y = 2^0 - 3 = 1 - 3 = -2 \).
- Since \( 2^x \) keeps increasing, the range of this function is \( y \geq -3 \).
Therefore, among the provided options, the function that has a range of \( y < 3 \) is:
[tex]\[ y = -(2)^x + 3. \][/tex]
1. \( y = 3 \cdot (2)^x \)
- This function involves an exponential term with a positive base.
- As \( x \) increases, \( 2^x \) increases, and hence \( 3 \cdot (2)^x \) also increases.
- Exponential functions like \( 2^x \) are always positive and increasing for all real \( x \).
- Therefore, the range of this function is \( y > 0 \).
2. \( y = 2 \cdot (3)^x \)
- Similarly, this is an exponential function with a positive base.
- As \( x \) increases, \( 3^x \) increases, and \( 2 \cdot (3)^x \) also increases.
- Since \( 3^x \) is always positive and increasing, the range of this function is also \( y > 0 \).
3. \( y = -(2)^x + 3 \)
- Here, the function involves a negative exponential term added to 3.
- As \( x \) increases, \( 2^x \) increases, but since it is multiplied by \(-1\), the value \(-(2)^x\) becomes more negative.
- Therefore, the function can take values less than 3 because as \( x \) increases, the negative term will make \( y \) decrease.
- This implies the range of this function can include values less than 3. Specifically, since the maximum value \( y \) can achieve is 3 (when \( x = 0 \)), and it decreases from there, the range of this function is \( y < 3 \).
4. \( y = (2)^x - 3 \)
- This function involves a positive exponential term subtracted by 3.
- As \( x \) increases, \( 2^x \) increases, and hence \( (2)^x - 3 \) also increases.
- The minimum value this function can take is when \( x = 0 \), which gives \( y = 2^0 - 3 = 1 - 3 = -2 \).
- Since \( 2^x \) keeps increasing, the range of this function is \( y \geq -3 \).
Therefore, among the provided options, the function that has a range of \( y < 3 \) is:
[tex]\[ y = -(2)^x + 3. \][/tex]
