The picture shows a black straight line with a red line segment $AB$ on it. This segment represents the graph of a certain function $f$ with the domain $[ 1.2; 3.2] $. Find the range of the function $f$.
Milan and Peter were tasked with solving the problem. They first tried to find the equation of a linear function $y = ax+ b$, whose graph is the black straight line pictured above. Each of them used a different approach.
Peter obtained the value of the coefficient $b = 0.2$ from the graph. Then, by substituting $y = 0$ and $x=-0.8$ into the slope-intercept form equation of a line $y = ax+ b$, he proceeded with the following calculation: $$ \begin{gather} 0 = -0.8a + 0.2 \cr a = 0.25 \end{gather} $$ He stated, that the sought linear function corresponding to the black line has the equation: $$ y = 0.25x+ 0.2 $$
Milan was convinced that if we know the intercepts of a line with the coordinate axes, we can write the equation of the line in intercept form immediately: $$ \frac{x}{- \frac45}+ \frac{y}{\frac15}= 1 $$ From this equation, Milan expressed $y$ in the following steps: $$ \begin{gather} \frac{5x}{-4} + \frac{5y}{1} = 1 \cr 5x- 20y = -4 \cr -20y = -4 - 5x \cr y = \frac15 + \frac{x}{4} \end{gather} $$ Peter and Milan were glad to arrive at the same equation for the black line and agreed on solving the rest of the problem together.
First, they decided to determine the coordinates of point $A$. Since they knew the $x$-coordinate of this point from the domain, they calculated the corresponding $y$-coordinate by substituting it into the obtained equation as follows: $$ \begin{gather} y = \frac15 + \frac{x}{4} = \frac15 + \frac{1.2}{4} = \frac15 + 0.3 = 0.5 \cr A= [1.2; 0.5] \end{gather} $$ Then, using the same approach, they found the coordinates of point $B$ as well: $$ \begin{gather} y = \frac15 + \frac{x}{4} = \frac15 + \frac{3.2}{4} = \frac15 + \frac45 = 1 \cr B= [3.2; 1] \end{gather} $$ Finally, they both concluded that the range of the function $f$, whose graph is the line segment marked in red in the picture, is $[ 0.5; 1]$.
Did Milan or Peter make a mistake?
No. They both proceeded correctly in their individual calculations as well as in their common ones.
Yes. They each got a different function corresponding to the black line.
Yes. They made a mistake in their common part. The linear function corresponding to the black line should have the equation $y = ax$.
Yes. Milan made a mistake. His procedure is incorrect but by chance he got the correct solution.
Yes. Both students proceeded incorrectly from the very beginning. The given data is not sufficient to find the range of the function.
