Exploring Linear Functions

By: Diana Brown

Explore different pairs of linear functions f(x) and g(x) and their graphs for:

i. h(x) = f(x) + g(x)

ii. h(x) = f(x).g(x)

iii. h(x) = f(x)/g(x)

iv. h(x) = f(g(x))

First lets choose two linear functions:

f(x) = 2x + 1

g(x) = 3x  4

Below are the graphs of the two functions:

Now lets explore the following graphs:

i. h(x) = f(x) + g(x) = (2x +1) + (3x  4) = 5x  3

Notice the graph is a linear function

ii. h(x) = f(x).g(x) = (2x +1)  (3x  4) = 6x²  5x  4

Notice the graph is a quadratic function

iii. h(x) = f(x)/g(x) = (2x +1) / (3x  4)

Notice the graph is a rational function

iv. h(x) = f(g(x)) = 2(3x  4) + 1 = 6x  7

Notice the graph is a linear function.

Lets try another pair of linear functions:

f(x) = - 3x + 5

g(x) = 1/3 x  2

See the graphs below for:

i. h(x) = (- 3x + 5) + (1/3 x  2)

ii. h(x) =  (- 3x + 5)  (1/3 x  2)

iii. h(x) =  (- 3x + 5) / (1/3 x  2)

iv. h(x) =  - 3(1/3 x  2) + 5

You will notice the graphs are also linear (i), quadratic (ii), rational (iii), and linear again (iv)

Lets try one more pair of linear functions:

f(x) = -2x  1

g(x) = -x + 3

View the graphs below for:

i. h(x) = (- 2x + 5) + (-x + 3)

ii. h(x) =  (- 2x  1)  (-x + 3)

iii. h(x) =  (- 2x  1) / (-x + 3)

iv. h(x) =  - 2(-x + 3)  1

As we can see from the graphs of each pair of linear functions, when you add two linear functions you get a new linear function; when you multiply two linear functions you get a quadratic function, when you divide two linear functions you get a rational function, and when you find the composite of two linear functions you get another linear functions.

If you take two set of linear functions and let the coefficients and constants be a, b, c, and d for any negative or positive real number, you can see the operations below:

Let f(x)=ax + b and g(x)=cx + d

i. h(x) = f(x) + g(x) = (ax + b) + (cx + d) = (a+b)x + (b + d); let a + b = e which is also a constant and b + d = f which is also a constant so therefore(ax + b) + (cx + d) =ex + f which is a linear function

ii. h(x) = f(x).g(x) = (ax + b) (cx + d) = (ax)(cx) + adx + bcx + bd = acx² +  adx + bcx + bd; since ac, ad, bc, and bd are constants than this is a quadratic equation.

iii. h(x) = f(x)/g(x) = (ax + b) (cx + d), when you divide two linear functions together your result is a rational function.

iv. h(x) = f(g(x)) = a(cx + d) + b = acx + ad + b, since ad and b are constants then together added they are some constant h, and ac is also a constant (j), so a(cx + d) + b = jx + h, which is a linear function.