Lastly let's investigate h(x) = f(g(x)) when n = 0

f(x) = -3

g(x) = 5

h(x) = n(nx + 5) - 3, therefore h(x) = -3

 

Again we start out with straight lines.

 

However, what happens when we make n=6

f(x) = 6x - 3

g(x) = 6x + 5

h(x) = 6(6x +5) - 3

 

Again we get a straight line. Let's compare the slopes and y-intercept as we did in the first investigation

f(x) = 6x - 3

g(x) = 6x + 5

h(x) = 6(6x +5) - 3

h(x) = 36x + 30 - 3

h(x) = 36x + 27

Therefore, the slope is the product of the slopes of f(x) and g(x) and the y-intercept is the product of the slope of f(x) and the y-intercept of g(x) plus the y-intercept of f(x)

 

Will this stand true when we set n = 20

f(x) = 20x - 3

g(x) = 20x + 5

h(x) = 20(20x + 5) - 3

 

h(x) = 20(20x+5) - 3

h(x) = 400x + 100 - 3

h(x) = 400x + 197

with the slope equal to 400 and the y-intercept equal to 197

Will the composition always be a linear function? Let's look at the general case

f(x) = ax + b

g(x) = cx + d

h(x) = a(cx + d) + b

h(x) = (ac)x + ad + b

h(x) = (ac)x + (ad +b)

if

s = ac

t = (ad + b)

then we get

h(x) = sx + t

a linear function

 


RETURN TO EXPLORING LINEAR EQUATIONS