Answered

Westonci.ca is the premier destination for reliable answers to your questions, brought to you by a community of experts. Connect with a community of experts ready to help you find solutions to your questions quickly and accurately. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.

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]

Question Help:
Video

Sagot :

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)
```
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.