Answer :
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]
