Final answer:
The equation of the linear function perpendicular to the line 3x - 2y = 4 and intersects it at x = 8 is y = (-2/3)x + 40/3. None of the multiple-choice options provided is correct.
Step-by-step explanation:
The subject of this question involves finding an equation for a linear function that is perpendicular to a given line and intersects the line at a specific x-value. To solve this, the slope of the given line 3x - 2y = 4 must first be determined. Rearrange the equation to the slope-intercept form y = mx + b to find the slope (m). The line 3x - 2y = 4 can be rewritten as y = (3/2)x - 2. So, the slope of this line is 3/2.
A line that is perpendicular to this line would have a slope that is the negative reciprocal of 3/2, which is -2/3. The equation of the linear function would therefore have the form y = (-2/3)x + b. To find b, the y-intercept, we use the fact that the line intersects 3x - 2y = 4 at x = 8. Substituting this value of x into the original line's equation yields 3(8) - 2y = 4, which simplifies to y = 10. So when x = 8, y = 10 on both lines (the original and the perpendicular line).
Now we substitute x = 8 and y = 10 into the perpendicular line's equation y = (-2/3)x + b to find b. 10 = (-2/3)(8) + b, which simplifies to b = 10 + (16/3) = 40/3. Thus, the linear function in slope-intercept form is y = (-2/3)x + 40/3. To convert this to the form g(x) = mx + b, we simply replace y with g(x), hence g(x) = (-2/3)x + 40/3. Notably, none of the options given, a) through d), match this result, suggesting there may have been a miscalculation or typo in the student's list of answer choices.