Stephanie Henderson's Assignment #1:
Combining Linear Functions
In this exploration, we will be looking at various combinations of two linear graphs. Each pair will look at the effects of f(x)+g(x), f(x)*g(x), f(x)/g(x), and f(g(x)).
f(x)=2x+3 
f(x)+g(x)=(2x+3)+(x2)=3x+1 
g(x)=x2 
f(x)*g(x)=(2x+3)*(x2)=2x^{2}+x6 
f(x)/g(x)=(2x+3)/(x2) 

f(g(x))=2(x2)+3=2x1 
Our next set of linear functions both have negative slopes. f(x)+g(x) continues to have a negative slope, but f(g(x)) has a positive slope now. This is because of the two negative "x"s being multiplied together. (Remember: two negatives make a positive.) f(x)*g(x) is still a parabola, and f(x)/g(x) is still a hyperbola.
f(x)=x+1 
f(x)+g(x)=(x+1)+(2x1)=3x 
g(x)=2x1 
f(x)*g(x)=(x+1)*(2x1)=2x^{2}x1 
f(x)/g(x)=(x+1)/(2x1) 

f(g(x))=(2x1)+1=2x+2 
When we look at perpendicular lines, we get some new and different results. In this particular case, f(x)+g(x) and f(g(x)) are parallel. This won't always happen though. Yintercepts just happened to land in the right spot. What relationship do we see between our yintercepts? f(x)*g(x) continues to be a parabola, but f(x)/g(x) is no longer a hyperbola  it's a line. Will this always happen? Again, luck of the draw. Do we see the same relationship between our yintercepts as we did before?
f(x)=x/2+1 
f(x)+g(x)=(x/2+1)+(x2)=x/21 
g(x)=x2 
f(x)*g(x)=(x/2+1)*(x2)=(1/2)x^{2}2x2 
f(x)/g(x)=(x/2+1)/(x2) 

f(g(x))=(x2)/2+1=x/2 
Next, let's examine two parallel lines. One would think this would create some interesting results, as the perpendicular lines did, but we actually don't have anything unusual about this set. f(x)+g(x) and f(g(x)) are both lines; f(x)*g(x) is a parabola; and f(x)/g(x) is a parabola.
f(x)=3x2 
f(x)+g(x)=(3x2)+(3x+1)=6x1 
g(x)=3x+1 
f(x)*g(x)=(3x2)*(3x+1)=9x^{2}3x2 
f(x)/g(x)=(3x2)/(3x+1) 

f(g(x))=3(3x+1)2=9x+1 
f(x)=2 
f(x)+g(x)=2+1=1 
g(x)=1 
f(x)*g(x)=2(1)=2 
f(x)/g(x)=2/1=2 

f(g(x))=2 

g(f(x))=1 