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.

First
lets choose two linear functions:

**f(x) = 2x + 1**

**g(x) = 3x 4**

Below
are the graphs of the two functions:

Now
lets 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.

Notice the graph is a quadratic function

**
**iii.

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.

Lets
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)

Lets
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.