At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Connect with a community of experts ready to help you find accurate solutions to your questions quickly and efficiently. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To find the approximate solution to the equation [tex]\( f(x) = g(x) \)[/tex] using the given functions:
[tex]\[ f(x) = \frac{1}{4} x^3 + 2 x - 1 \][/tex]
[tex]\[ g(x) = 5^{(x-1)} - 3 \][/tex]
we need to create a table of values for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] from [tex]\( x = 1.0 \)[/tex] to [tex]\( x = 2.5 \)[/tex] in intervals of [tex]\( 0.25 \)[/tex]. Then we compare the values of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] to see when they are approximately equal. Here's the table:
| [tex]\( x \)[/tex] | [tex]\( f(x) \)[/tex] | [tex]\( g(x) \)[/tex] |
|---------|--------------------|---------------------------|
| 1.0 | 1.25 | -2.0 |
| 1.25 | 1.98828125 | -1.5046512187787795 |
| 1.5 | 2.84375 | -0.7639320225002102 |
| 1.75 | 3.83984375 | 0.34370152488211003 |
| 2.0 | 5.0 | 2.0 |
| 2.25 | 6.34765625 | 4.476743906106103 |
| 2.5 | 7.90625 | 8.180339887498949 |
To find an approximate solution to [tex]\( f(x) = g(x) \)[/tex], we look for the [tex]\( x \)[/tex]-value where [tex]\( f(x) \approx g(x) \)[/tex].
From the table, we observe the following differences:
- For [tex]\( x = 1.0 \)[/tex], [tex]\( f(x) = 1.25 \)[/tex] and [tex]\( g(x) = -2.0 \)[/tex] (difference is [tex]\( 3.25 \)[/tex]).
- For [tex]\( x = 1.25 \)[/tex], [tex]\( f(x) = 1.98828125 \)[/tex] and [tex]\( g(x) = -1.5046512187787795 \)[/tex] (difference is [tex]\( 3.4929324687787795 \)[/tex]).
- For [tex]\( x = 1.5 \)[/tex], [tex]\( f(x) = 2.84375 \)[/tex] and [tex]\( g(x) = -0.7639320225002102 \)[/tex] (difference is [tex]\( 3.6076820225002103 \)[/tex]).
- For [tex]\( x = 1.75 \)[/tex], [tex]\( f(x) = 3.83984375 \)[/tex] and [tex]\( g(x) = 0.34370152488211003 \)[/tex] (difference is [tex]\( 3.49614222511789 \)[/tex]).
- For [tex]\( x = 2.0 \)[/tex], [tex]\( f(x) = 5.0 \)[/tex] and [tex]\( g(x) = 2.0 \)[/tex] (difference is [tex]\( 3.0 \)[/tex]).
- For [tex]\( x = 2.25 \)[/tex], [tex]\( f(x) = 6.34765625 \)[/tex] and [tex]\( g(x) = 4.476743906106103 \)[/tex] (difference is [tex]\( 1.870912343893897 \)[/tex]).
- For [tex]\( x = 2.5 \)[/tex], [tex]\( f(x) = 7.90625 \)[/tex] and [tex]\( g(x) = 8.180339887498949 \)[/tex] (difference is [tex]\( 0.2740898874989492 \)[/tex]).
Examining the differences, none of the values result in [tex]\( f(x) \approx g(x) \)[/tex] to within 0.1.
Thus, there is no [tex]\( x \)[/tex]-value from the table where [tex]\( f(x) \)[/tex] approximates [tex]\( g(x) \)[/tex] closely enough to meet the criteria set in the question. Therefore, the correct answer is:
```
None of the given options (A, B, C, D) provide a value close enough to satisfy [tex]\( f(x) = g(x) \)[/tex] within the specified tolerance of 0.1. Given the choices, none is correct.
```
However, based on the fact that we need to pick from the given options and [tex]\( x = 1.75 \)[/tex] seems to be the closest value since [tex]\( f(1.75) \approx 3.83984375 \)[/tex] and [tex]\( g(1.75) \approx 0.34370152488211003 \)[/tex], purely based on inspection, although not technically correct within a narrow tolerance.
So, if forced to choose:
- Closest answer from provided options is:
A. [tex]\( x \approx 1.75 \)[/tex]
[tex]\[ f(x) = \frac{1}{4} x^3 + 2 x - 1 \][/tex]
[tex]\[ g(x) = 5^{(x-1)} - 3 \][/tex]
we need to create a table of values for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] from [tex]\( x = 1.0 \)[/tex] to [tex]\( x = 2.5 \)[/tex] in intervals of [tex]\( 0.25 \)[/tex]. Then we compare the values of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] to see when they are approximately equal. Here's the table:
| [tex]\( x \)[/tex] | [tex]\( f(x) \)[/tex] | [tex]\( g(x) \)[/tex] |
|---------|--------------------|---------------------------|
| 1.0 | 1.25 | -2.0 |
| 1.25 | 1.98828125 | -1.5046512187787795 |
| 1.5 | 2.84375 | -0.7639320225002102 |
| 1.75 | 3.83984375 | 0.34370152488211003 |
| 2.0 | 5.0 | 2.0 |
| 2.25 | 6.34765625 | 4.476743906106103 |
| 2.5 | 7.90625 | 8.180339887498949 |
To find an approximate solution to [tex]\( f(x) = g(x) \)[/tex], we look for the [tex]\( x \)[/tex]-value where [tex]\( f(x) \approx g(x) \)[/tex].
From the table, we observe the following differences:
- For [tex]\( x = 1.0 \)[/tex], [tex]\( f(x) = 1.25 \)[/tex] and [tex]\( g(x) = -2.0 \)[/tex] (difference is [tex]\( 3.25 \)[/tex]).
- For [tex]\( x = 1.25 \)[/tex], [tex]\( f(x) = 1.98828125 \)[/tex] and [tex]\( g(x) = -1.5046512187787795 \)[/tex] (difference is [tex]\( 3.4929324687787795 \)[/tex]).
- For [tex]\( x = 1.5 \)[/tex], [tex]\( f(x) = 2.84375 \)[/tex] and [tex]\( g(x) = -0.7639320225002102 \)[/tex] (difference is [tex]\( 3.6076820225002103 \)[/tex]).
- For [tex]\( x = 1.75 \)[/tex], [tex]\( f(x) = 3.83984375 \)[/tex] and [tex]\( g(x) = 0.34370152488211003 \)[/tex] (difference is [tex]\( 3.49614222511789 \)[/tex]).
- For [tex]\( x = 2.0 \)[/tex], [tex]\( f(x) = 5.0 \)[/tex] and [tex]\( g(x) = 2.0 \)[/tex] (difference is [tex]\( 3.0 \)[/tex]).
- For [tex]\( x = 2.25 \)[/tex], [tex]\( f(x) = 6.34765625 \)[/tex] and [tex]\( g(x) = 4.476743906106103 \)[/tex] (difference is [tex]\( 1.870912343893897 \)[/tex]).
- For [tex]\( x = 2.5 \)[/tex], [tex]\( f(x) = 7.90625 \)[/tex] and [tex]\( g(x) = 8.180339887498949 \)[/tex] (difference is [tex]\( 0.2740898874989492 \)[/tex]).
Examining the differences, none of the values result in [tex]\( f(x) \approx g(x) \)[/tex] to within 0.1.
Thus, there is no [tex]\( x \)[/tex]-value from the table where [tex]\( f(x) \)[/tex] approximates [tex]\( g(x) \)[/tex] closely enough to meet the criteria set in the question. Therefore, the correct answer is:
```
None of the given options (A, B, C, D) provide a value close enough to satisfy [tex]\( f(x) = g(x) \)[/tex] within the specified tolerance of 0.1. Given the choices, none is correct.
```
However, based on the fact that we need to pick from the given options and [tex]\( x = 1.75 \)[/tex] seems to be the closest value since [tex]\( f(1.75) \approx 3.83984375 \)[/tex] and [tex]\( g(1.75) \approx 0.34370152488211003 \)[/tex], purely based on inspection, although not technically correct within a narrow tolerance.
So, if forced to choose:
- Closest answer from provided options is:
A. [tex]\( x \approx 1.75 \)[/tex]
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.