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Question 10

The graph of the function [tex]f(x)=e^{-x}-9[/tex] can be obtained from the graph of [tex]g(x)=e^x[/tex] by one of the following actions:

(a) Reflecting the graph of [tex]g(x)[/tex] in the [tex]x[/tex]-axis;
(b) Reflecting the graph of [tex]g(x)[/tex] in the [tex]y[/tex]-axis;

Your answer is (input a or b): [tex]$\square$[/tex]

Then, by one of the following actions:

(a) Shifting the resulting graph to the right 9 units;
(b) Shifting the resulting graph to the left 9 units;
(c) Shifting the resulting graph upward 9 units;
(d) Shifting the resulting graph downward 9 units;

Your answer is (input a, b, c, or d): [tex]$\square$[/tex]

Is the domain of the function [tex]f(x)[/tex] still [tex](-\infty, \infty)[/tex]?
Your answer is (input Yes or No): [tex]$\square$[/tex]

The range of the function [tex]f(x)[/tex] is [tex](A, \infty)[/tex], the value of [tex]A[/tex] is: [tex]$\square$[/tex]

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GinnyAnswer
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)
```
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