Answer :
To obtain the graph of the function [tex]\( f(x)=e^{-x}-9 \)[/tex] from the graph of [tex]\( g(x)=e^x \)[/tex], let's follow a detailed step-by-step transformation process:
1. Reflection in the y-axis:
- Consider the original function [tex]\( g(x) = e^x \)[/tex].
- Reflecting the graph of [tex]\( g(x) \)[/tex] in the y-axis involves replacing [tex]\( x \)[/tex] with [tex]\( -x \)[/tex].
- Therefore, the new function after reflection would be [tex]\( g(-x) = e^{-x} \)[/tex].
- Thus, choice (b) reflecting the graph of [tex]\( g(x) \)[/tex] in the y-axis is correct.
2. Shifting the resulting graph:
- After reflecting the graph in the y-axis, we obtained [tex]\( g(-x) = e^{-x} \)[/tex].
- To obtain [tex]\( f(x) = e^{-x} - 9 \)[/tex], we now need to shift this reflected graph downward by 9 units. This means subtracting 9 from [tex]\( e^{-x} \)[/tex].
- Therefore, choice (d) shifting the resulting graph downward 9 units is correct.
3. Domain of [tex]\( f(x) \)[/tex]:
- The domain of the function [tex]\( g(x) = e^x \)[/tex] is all real numbers, [tex]\((-\infty, \infty)\)[/tex].
- Reflecting in the y-axis and then shifting downward does not change the domain of the function.
- Therefore, the domain of [tex]\( f(x) = e^{-x} - 9 \)[/tex] is still [tex]\((-\infty, \infty)\)[/tex].
- The answer is Yes.
4. Range of [tex]\( f(x) \)[/tex]:
- For [tex]\( g(x) = e^x \)[/tex], the range is [tex]\((0, \infty)\)[/tex] because [tex]\( e^x \)[/tex] is always positive.
- When we reflect it in the y-axis to get [tex]\( e^{-x} \)[/tex], the range remains [tex]\((0, \infty)\)[/tex].
- Shifting this downward by 9 units, we subtract 9 from each value in the range. So the range of [tex]\( e^{-x} - 9 \)[/tex] is [tex]\((-9, \infty)\)[/tex].
- Therefore, the value of [tex]\( A \)[/tex] is [tex]\( -9 \)[/tex].
Summarizing, the answers are:
- Reflecting the graph of [tex]\( g(x) \)[/tex] in the y-axis: b
- Shifting the resulting graph downward 9 units: d
- The domain of [tex]\( f(x) \)[/tex] is still [tex]\((-\infty, \infty)\)[/tex]: Yes
- The range of the function [tex]\( f(x) = e^{-x} - 9 \)[/tex] is [tex]\((-9, \infty)\)[/tex] and the value of [tex]\( A \)[/tex] is -9.
Thus, the final answers are:
```
(b, d, Yes, -9)
```
1. Reflection in the y-axis:
- Consider the original function [tex]\( g(x) = e^x \)[/tex].
- Reflecting the graph of [tex]\( g(x) \)[/tex] in the y-axis involves replacing [tex]\( x \)[/tex] with [tex]\( -x \)[/tex].
- Therefore, the new function after reflection would be [tex]\( g(-x) = e^{-x} \)[/tex].
- Thus, choice (b) reflecting the graph of [tex]\( g(x) \)[/tex] in the y-axis is correct.
2. Shifting the resulting graph:
- After reflecting the graph in the y-axis, we obtained [tex]\( g(-x) = e^{-x} \)[/tex].
- To obtain [tex]\( f(x) = e^{-x} - 9 \)[/tex], we now need to shift this reflected graph downward by 9 units. This means subtracting 9 from [tex]\( e^{-x} \)[/tex].
- Therefore, choice (d) shifting the resulting graph downward 9 units is correct.
3. Domain of [tex]\( f(x) \)[/tex]:
- The domain of the function [tex]\( g(x) = e^x \)[/tex] is all real numbers, [tex]\((-\infty, \infty)\)[/tex].
- Reflecting in the y-axis and then shifting downward does not change the domain of the function.
- Therefore, the domain of [tex]\( f(x) = e^{-x} - 9 \)[/tex] is still [tex]\((-\infty, \infty)\)[/tex].
- The answer is Yes.
4. Range of [tex]\( f(x) \)[/tex]:
- For [tex]\( g(x) = e^x \)[/tex], the range is [tex]\((0, \infty)\)[/tex] because [tex]\( e^x \)[/tex] is always positive.
- When we reflect it in the y-axis to get [tex]\( e^{-x} \)[/tex], the range remains [tex]\((0, \infty)\)[/tex].
- Shifting this downward by 9 units, we subtract 9 from each value in the range. So the range of [tex]\( e^{-x} - 9 \)[/tex] is [tex]\((-9, \infty)\)[/tex].
- Therefore, the value of [tex]\( A \)[/tex] is [tex]\( -9 \)[/tex].
Summarizing, the answers are:
- Reflecting the graph of [tex]\( g(x) \)[/tex] in the y-axis: b
- Shifting the resulting graph downward 9 units: d
- The domain of [tex]\( f(x) \)[/tex] is still [tex]\((-\infty, \infty)\)[/tex]: Yes
- The range of the function [tex]\( f(x) = e^{-x} - 9 \)[/tex] is [tex]\((-9, \infty)\)[/tex] and the value of [tex]\( A \)[/tex] is -9.
Thus, the final answers are:
```
(b, d, Yes, -9)
```
