Given the quadratic function $f(x) = x^2 − 2x$, find the real number $k$ for which the equation $$ f(3) = k\cdot f\left(\frac13\right) $$ holds.
Charles solved the task as follows:
1) First, he calculated the value of the function at $x=3$: $$ f(3) = 3^2 − 2\cdot 3 = 9 − 6 = 3 $$
2) Similarly, he calculated the value of the function at $x=\frac13$: $$ f\left(\frac13\right) = \left(\frac13\right)^2− 2\cdot \frac13 = \frac19 − \frac23 = \frac19 − \frac49 = − \frac39 = − \frac13 $$
3) After that, he substituted the calculated values into the given equation: $$ f(3) = k\cdot f\left(\frac13\right) $$ and got: $$ 3 = k\cdot \left(− \frac13\right) $$
4) The solution to this equation is: $$ k = −9 $$
Is Charles’s solution correct? If no, identify a mistake.
No. There is a mistake in step (1). There is a numerical error.
No. There is a mistake in step (2). There is a numerical error.
No. There is a mistake in step (3). It should have been: $$ 3 = k − \frac13 $$
No. There is a mistake in step (4). The correct solution to the equation should be: $$ k = − \frac19 $$
Yes. There is no mistake.
There is a numerical error in step (2). The correct calculation is: $$ f\left(\frac13\right) = \left(\frac13\right)^2− 2\cdot \frac13 = \frac19 − \frac23 = \frac19 − \frac69 = − \frac59 $$
By substituting into the given equation, we get: $$ \begin{gather} f(3) = k\cdot f\left(\frac13\right) \cr 3 = k\cdot \left(− \frac59\right) \cr k = − \frac{27}{5} \end{gather} $$