Question

Draw the graph and find domain and range
f(x) =
${x }^{2} + 4 \\ (x - 3) {}^{2} \\ x {}^{2} + 2x + 3$
​

Answer 1

Clearly the function f is defined for all real numbers and so its domain is R.

Also note that f(x) >/=0 for all x, and given any non-negative real number y, taking x =y+1 we see that f(x)=||y+1|-1| =|y+1–1|=|y|=y, because |y+1|=y+1, as (y+1)>0, as y>/=0, and for the same reason |y|=y. Hence every real number y>/=0 is f(y+1). Therefore the range of f is {y € R : y>/=0}.

As for drawing the graph of the function f, for x>/=1, f(x)=x-1. For 0 =x<1, f(x)=1-x. Observe that the function is an even function of x, as f(-x)=f(x) for all x. Hence for negative values of x, the graph can be obtained by reflecting it about the y-axis. So for -1 < x < 0,

f(x) =f(-x)=1-(-x) (as 0<(-x)<1) = 1+x, and finally for x=(-1), f(x) = f(-x) =-(-x)-1 (since (-x)>/=1)

= -(x+1). Thus the graph looks like a gigantic W, being given by

1: x =(-1) : f(x)=-(x+1)

2:(-1) < x = 0 : f(x)=(x+1)

3: 0 < x < 1 : f(x)= (1-x)

4: x >/= 1 : f(x) = (x-1). The angular points of this giant W are at (-1,0), (0,1) and (1,0).