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Sagot :
To determine which of the given functions have a domain of [tex]\((-\infty, \infty)\)[/tex] and a range of [tex]\((-\infty, 4)\)[/tex], we'll analyze each one step by step.
Recall that all exponential functions of the form [tex]\( a^x \)[/tex] have a domain of [tex]\((-\infty, \infty)\)[/tex]. However, we need to also consider how these forms transform when manipulated as described below.
### Function Analysis
1. A. [tex]\( v = -(3)^x - 4 \)[/tex]
- This function is a transformation of the exponential function [tex]\( 3^x \)[/tex].
- The original exponential function [tex]\( 3^x \)[/tex] has a range of [tex]\( (0, \infty) \)[/tex].
- By applying the negative sign: [tex]\( -(3)^x \)[/tex] changes the range to [tex]\( (-\infty, 0) \)[/tex] because the exponential growth is reflected downwards.
- Subtracting 4 from all values: [tex]\( -(3)^x - 4 \)[/tex] shifts the entire range down by 4 units, changing the range to [tex]\( (-\infty, -4) \)[/tex].
- Conclusion: The range does not match [tex]\((-\infty, 4)\)[/tex]. Thus, this function is not an example.
2. B. [tex]\( v = -(3)^x + 4 \)[/tex]
- Starting with the exponential function [tex]\( 3^x \)[/tex] which has a range of [tex]\( (0, \infty) \)[/tex].
- Applying the negative sign: [tex]\( -(3)^x \)[/tex] changes the range to [tex]\( (-\infty, 0) \)[/tex].
- Adding 4 to all values: [tex]\( -(3)^x + 4 \)[/tex] shifts the entire range up by 4 units, changing the range to [tex]\( (-\infty, 4) \)[/tex].
- Conclusion: This range matches [tex]\((-\infty, 4)\)[/tex]. Thus, this function is an example.
3. C. [tex]\( y = -(0.25)^x + 4 \)[/tex]
- The function [tex]\( 0.25^x \)[/tex] is an exponential function with a range of [tex]\( (0, \infty) \)[/tex].
- Applying the negative sign: [tex]\( -(0.25)^x \)[/tex] transforms the range to [tex]\( (-\infty, 0) \)[/tex].
- Adding 4 to all values: [tex]\( -(0.25)^x + 4 \)[/tex] shifts the entire range up by 4 units, changing the range to [tex]\( (-\infty, 4) \)[/tex].
- Conclusion: This range matches [tex]\((-\infty, 4)\)[/tex]. Thus, this function is an example.
4. D. [tex]\( y = -(0.25)^x - 4 \)[/tex]
- Consider the function [tex]\( 0.25^x \)[/tex], which has a range of [tex]\( (0, \infty) \)[/tex].
- Applying the negative sign: [tex]\( -(0.25)^x \)[/tex] changes the range to [tex]\( (-\infty, 0) \)[/tex].
- Subtracting 4 from all values: [tex]\( -(0.25)^x - 4 \)[/tex] shifts the entire range down by 4 units, changing the range to [tex]\( (-\infty, -4) \)[/tex].
- Conclusion: The range does not match [tex]\((-\infty, 4)\)[/tex]. Thus, this function is not an example.
### Summary
The functions that have a domain of [tex]\((-\infty, \infty)\)[/tex] and a range of [tex]\((-\infty, 4)\)[/tex] are:
- B. [tex]\( v = -(3)^x + 4 \)[/tex]
- C. [tex]\( y = -(0.25)^x + 4 \)[/tex]
Recall that all exponential functions of the form [tex]\( a^x \)[/tex] have a domain of [tex]\((-\infty, \infty)\)[/tex]. However, we need to also consider how these forms transform when manipulated as described below.
### Function Analysis
1. A. [tex]\( v = -(3)^x - 4 \)[/tex]
- This function is a transformation of the exponential function [tex]\( 3^x \)[/tex].
- The original exponential function [tex]\( 3^x \)[/tex] has a range of [tex]\( (0, \infty) \)[/tex].
- By applying the negative sign: [tex]\( -(3)^x \)[/tex] changes the range to [tex]\( (-\infty, 0) \)[/tex] because the exponential growth is reflected downwards.
- Subtracting 4 from all values: [tex]\( -(3)^x - 4 \)[/tex] shifts the entire range down by 4 units, changing the range to [tex]\( (-\infty, -4) \)[/tex].
- Conclusion: The range does not match [tex]\((-\infty, 4)\)[/tex]. Thus, this function is not an example.
2. B. [tex]\( v = -(3)^x + 4 \)[/tex]
- Starting with the exponential function [tex]\( 3^x \)[/tex] which has a range of [tex]\( (0, \infty) \)[/tex].
- Applying the negative sign: [tex]\( -(3)^x \)[/tex] changes the range to [tex]\( (-\infty, 0) \)[/tex].
- Adding 4 to all values: [tex]\( -(3)^x + 4 \)[/tex] shifts the entire range up by 4 units, changing the range to [tex]\( (-\infty, 4) \)[/tex].
- Conclusion: This range matches [tex]\((-\infty, 4)\)[/tex]. Thus, this function is an example.
3. C. [tex]\( y = -(0.25)^x + 4 \)[/tex]
- The function [tex]\( 0.25^x \)[/tex] is an exponential function with a range of [tex]\( (0, \infty) \)[/tex].
- Applying the negative sign: [tex]\( -(0.25)^x \)[/tex] transforms the range to [tex]\( (-\infty, 0) \)[/tex].
- Adding 4 to all values: [tex]\( -(0.25)^x + 4 \)[/tex] shifts the entire range up by 4 units, changing the range to [tex]\( (-\infty, 4) \)[/tex].
- Conclusion: This range matches [tex]\((-\infty, 4)\)[/tex]. Thus, this function is an example.
4. D. [tex]\( y = -(0.25)^x - 4 \)[/tex]
- Consider the function [tex]\( 0.25^x \)[/tex], which has a range of [tex]\( (0, \infty) \)[/tex].
- Applying the negative sign: [tex]\( -(0.25)^x \)[/tex] changes the range to [tex]\( (-\infty, 0) \)[/tex].
- Subtracting 4 from all values: [tex]\( -(0.25)^x - 4 \)[/tex] shifts the entire range down by 4 units, changing the range to [tex]\( (-\infty, -4) \)[/tex].
- Conclusion: The range does not match [tex]\((-\infty, 4)\)[/tex]. Thus, this function is not an example.
### Summary
The functions that have a domain of [tex]\((-\infty, \infty)\)[/tex] and a range of [tex]\((-\infty, 4)\)[/tex] are:
- B. [tex]\( v = -(3)^x + 4 \)[/tex]
- C. [tex]\( y = -(0.25)^x + 4 \)[/tex]
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