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Sagot :
To determine which function has a range of [tex]\( y < 3 \)[/tex], we need to analyze each function carefully.
1. Function [tex]\( y = 3 \cdot 2^x \)[/tex]:
This is an exponential function where the base is 2 and the coefficient is 3.
- As [tex]\( x \)[/tex] increases, [tex]\( 2^x \)[/tex] grows exponentially.
- Multiplying [tex]\( 2^x \)[/tex] by 3 does not change the exponential growth nature; it just scales it.
- Therefore, the range of this function is [tex]\( y > 0 \)[/tex], since [tex]\( y \)[/tex] will grow indefinitely as [tex]\( x \)[/tex] increases.
- This function's range is not restricted to values below 3.
2. Function [tex]\( y = 2 \cdot 3^x \)[/tex]:
This is another exponential function, but with a base of 3 and a coefficient of 2.
- Similar to the first function, as [tex]\( x \)[/tex] increases, [tex]\( 3^x \)[/tex] grows exponentially.
- Multiplying [tex]\( 3^x \)[/tex] by 2 scales the function but retains its exponential growth.
- Hence, the range of this function is also [tex]\( y > 0 \)[/tex].
- This function's range is not restricted to values below 3.
3. Function [tex]\( y = -(2^x) + 3 \)[/tex]:
This function involves an exponential term with a negative sign and a constant term added.
- As [tex]\( x \)[/tex] increases, [tex]\( 2^x \)[/tex] still grows exponentially.
- The negative sign indicates that [tex]\( 2^x \)[/tex] is subtracted from 3.
- This means, as [tex]\( x \)[/tex] becomes very large, [tex]\( -(2^x) \)[/tex] becomes a very large negative number.
- Therefore, [tex]\( y \)[/tex] approaches 3 from below (since [tex]\( y = 3 - 2^x \)[/tex]).
- Thus, this function's range is indeed [tex]\( y < 3 \)[/tex].
4. Function [tex]\( y = 2^x - 3 \)[/tex]:
This is an exponential function similar to the first two, but it is shifted down by 3 units.
- As [tex]\( x \)[/tex] increases, [tex]\( 2^x \)[/tex] grows exponentially.
- Subtracting 3 shifts the entire graph down by 3 units.
- Therefore, the range of this function is [tex]\( y > -3 \)[/tex], which means [tex]\( y \)[/tex] can definitely be greater than 3 at larger [tex]\( x \)[/tex].
- This function's range is not restricted to values below 3.
From this analysis, the function with a range of [tex]\( y < 3 \)[/tex] is:
[tex]\[ y = -(2^x) + 3 \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{3} \][/tex]
1. Function [tex]\( y = 3 \cdot 2^x \)[/tex]:
This is an exponential function where the base is 2 and the coefficient is 3.
- As [tex]\( x \)[/tex] increases, [tex]\( 2^x \)[/tex] grows exponentially.
- Multiplying [tex]\( 2^x \)[/tex] by 3 does not change the exponential growth nature; it just scales it.
- Therefore, the range of this function is [tex]\( y > 0 \)[/tex], since [tex]\( y \)[/tex] will grow indefinitely as [tex]\( x \)[/tex] increases.
- This function's range is not restricted to values below 3.
2. Function [tex]\( y = 2 \cdot 3^x \)[/tex]:
This is another exponential function, but with a base of 3 and a coefficient of 2.
- Similar to the first function, as [tex]\( x \)[/tex] increases, [tex]\( 3^x \)[/tex] grows exponentially.
- Multiplying [tex]\( 3^x \)[/tex] by 2 scales the function but retains its exponential growth.
- Hence, the range of this function is also [tex]\( y > 0 \)[/tex].
- This function's range is not restricted to values below 3.
3. Function [tex]\( y = -(2^x) + 3 \)[/tex]:
This function involves an exponential term with a negative sign and a constant term added.
- As [tex]\( x \)[/tex] increases, [tex]\( 2^x \)[/tex] still grows exponentially.
- The negative sign indicates that [tex]\( 2^x \)[/tex] is subtracted from 3.
- This means, as [tex]\( x \)[/tex] becomes very large, [tex]\( -(2^x) \)[/tex] becomes a very large negative number.
- Therefore, [tex]\( y \)[/tex] approaches 3 from below (since [tex]\( y = 3 - 2^x \)[/tex]).
- Thus, this function's range is indeed [tex]\( y < 3 \)[/tex].
4. Function [tex]\( y = 2^x - 3 \)[/tex]:
This is an exponential function similar to the first two, but it is shifted down by 3 units.
- As [tex]\( x \)[/tex] increases, [tex]\( 2^x \)[/tex] grows exponentially.
- Subtracting 3 shifts the entire graph down by 3 units.
- Therefore, the range of this function is [tex]\( y > -3 \)[/tex], which means [tex]\( y \)[/tex] can definitely be greater than 3 at larger [tex]\( x \)[/tex].
- This function's range is not restricted to values below 3.
From this analysis, the function with a range of [tex]\( y < 3 \)[/tex] is:
[tex]\[ y = -(2^x) + 3 \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{3} \][/tex]
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