Another HSC Q

[AlonzoN]

so then would x have to be <= 0? or the other way around?

 

[Harry_the_cat]

If the domain of g is x<=0, what is the range?

 

[Harry_the_cat]

We just have to look at the graph to the left of x=0.

 

[AlonzoN]

y >= 0?

 

[Harry_the_cat]

Why not y>=-1

 

[AlonzoN]

cause we don't want it to be below the x-axis?

 

[Harry_the_cat]

Yeah we don't. But at the moment, just consider the part of your graph where x<=0. What is the range of the y-values? -1 and up , right?

 

[AlonzoN]

yep.

 

[Harry_the_cat]

So c isn't 0. Remember we are trying to find the largest value of c so that the range of g is a subset of >=0.
So what if we now consider if x<= -1, what is the range? So just consider the part of your graph to the left of -1. Is the range still -1 and above?

 

[Harry_the_cat]

Sorry just edited. I meant x<= -1.

 

[Harry_the_cat]

Then consider x<= -2.
Then consider x<= -3.

 

[AlonzoN]

if x <= -1, y>= 0
if x<= -2, y >= -1
if x<= -3, y >=0

 

[Harry_the_cat]

If x<= -1, the range is still y>= -1. You shpuld be looking at the part of you graph to the left of -1. That still includes the TP.

 

[AlonzoN]

oh.

 

[Harry_the_cat]

Do you agree?

 

[Harry_the_cat]

if x <= -1, y>= 0
if x<= -2, y >= -1
if x<= -3, y >=0

Your second and third answers are correct.

 

[Harry_the_cat]

Can you see that it's not until you look at x<= -3, that you have a range of y>=0 ?

 

[AlonzoN]

yes

 

[Harry_the_cat]

So c must be -3.
That is if x<= -3, then the range of g(x) is a subset of the domain of f(x).

 

[Harry_the_cat]

Of course, if x<= -4, then the range of g would also be a (smaller) subset of f. That is why the question asks for the LARGEST value of c.