Answer
To remove the discontinuity at $x=1$, $f(1)$ should be changed to $2$.
Work Step by Step
To remove the discontinuity, $f(1)$ must acquire a value so that $\lim_{x\to1}f(x)=f(1)$. In other words, $f(1)$ needs to be changed to the same value as $\lim_{x\to1}f(x)$.
For value of $\lim_{x\to1}f(x)$, you can always refer back to Exercise 6. I would calculate $\lim_{x\to1}f(x)$ again here though.
- As $x\to1^-$, since $x\lt1$, we employ the function $f(x)=2x$
$$\lim_{x\to1^-}f(x)=\lim_{x\to1^-}(2x)=2\times1=2$$
- As $x\to1^+$, since $x\gt1$, we employ the function $f(x)=-2x+4$
$$\lim_{x\to1^+}f(x)=\lim_{x\to1^+}(-2x+4)=-2\times1+4=2$$
Therefore, $\lim_{x\to1}f(x)=\lim_{x\to1^+}f(x)=\lim_{x\to1^-}f(x)=2$
So, to remove the discontinuity at $x=1$, $f(1)$ should be changed to $2$.
