Graphing functions on a graph

Question

Need help solving this problem step by step!

Answer 1

The piece-wise function is something that is asking,"What acts like $f(x)=x$ up to and less than $1$, and acts like $f(x)=2-x^2$ afterwards? So what (single) graph doesn't act like $f(x)=x$ up to $1$?

The next thing to realize is that for large enough values of $x$, $2-x^2<0$, so let's ask what (single) graph (left over) doesn't do this?

Now you should have two graphs left. Now with the same line of reasoning, you should realize that once the graph goes negative it should actually stay negative (i.e. once $2-x^2 <0$ for some $x$ it will stay negative because the $-x^2$ just makes it "more negative". Which of the two left doesn't do this?

Now you're left with the answer!

Answer 2

The graph of $f(x)=x$ is:

So,we can reject the answer $B$.

$2-x^2$ is a parabola..$2-x^2$ is always negative for $x>1$,for example,for $x=2$, $2-x^2=-2<0$.

So,we can reject the answer $D$.

Also, for $x>1$, $f'(x)=-2x<0 \forall x>1$,so $f$ is decreasing for $x>1$.So,the right answer is $C$,since at graph $A$,the function is increasing for $x>2$.