**Assignment 1 by Butila Kialeuka(Diekumpuna)**

__Exploring two linear functions through
compositions.__

The purpose of the assignment is to explore with different
pairs of f(x) and g(x) along with their graphs in order to analyze the results
of various operations including: addition, subtraction, multiplication and division
of the two functions.

**Set 1 y=
f(x) = 2x + 1 y=g(x)
= -3x – 5**

Analyzing the compositions of the above linear functions,
each operation yields a different result respectively as the two equations have
different slopes. When adding and subtracting the two linear functions, the
result remains linear where as when multiplying or dividing the two functions,
the result is completely different as expected. The product of the two linear
functions results in a parabola as it becomes a quadratic functions. The
quotient of the linear functions, on the other hand, results in a
hyperbola. It is important to note
that all of the compositions remain functions as they all pass the vertical
test respectively. The composition of f(g(x)) will always yield a linear
relation since the coefficient of
f(x) is being multiplied to both addends of g(x ). When conducting the
mathematics manually, the two functions resulted in linear functions as well in
exception of when the two were being divided and multiplied. Product confirmed
the graph and resulted into a parabola where as the quotient resulted into a
hyperbola.

**Set 2 y=
f(x) = 2x + 1 y=
g(x) = 2x -3**

Using another pair of linear functions having the same slope,
which are parallel lines, the same
result was generated as it was in the first set of equations,
in exception of when finding the difference of the two linear functions
which yield a line with zero slope as the group of variables cancel each other
out. These are still functions as the vertical line test was conducted to
verify the validity of the function. The composition of f(g(x)) will always
yield a linear relation since the coefficient of f(x) is being multiplied to both addends of g(x ). The
manual manipulation of the mathematics resulted in responses support the
graphical representations of the functions.

**Set 3 y=f(x)
= 2x + 1 y=g(x)
= -x****∕2 +4**

The third set involves the linear functions that are perpendicular
(negative reciprocal of each other).Both the difference and sum of the
composite intersect with the second function, g(x). The product of the two
functions remains a parabola where as the quotient results in a hyperbola. All
of the compositions remain functions as they are validated through the vertical
line test. These assumptions can be made provided the x remains the independent
variable. The composition of f(g(x)) will always yield a linear relation since
the coefficient of f(x) is being multiplied
to both addends of g(x ). The manual manipulation of the mathematics resulted
in responses support the graphical representations of the functions.