(a) Find an equation for the family of linear functions with slope 2 and several members of the family.(a) Find an equation for the family of...

a) We'll use the slope- intercept form of the equation of
linear function:

y = mx + n, where m is the slope and n is
the y intercept of the line.

In the given case, when the
slope m = 2, we can write the equation:

y = 2x +
n

To identify several members of this family, we'll plug in
values for n:

n = 1 => y = 2x +
1

n = 2 => y = 2x + 2

n
= `sqrt(2)` => y = 2x + `sqrt(2)`

Therefore, the
number of members of the family of linear functions whose slope is m=2 is infinite,
because the number of real values of n is infinite, too.

b)
To find an equation of a linear function, respecting the given condition f(2) = 1, we'll
recall the form of a linear function:

f(x) = mx +
n

If x = 2 => f(2) = 2m +
n

If f(2) = 1 => 2m + n = 1 => m = (1 -
n)/2

We'll write the linear function in terms of x and
n:

f(x) = (1-n)x/2 +
n

Therefore, the number of members of the family of linear
function f(x) = (1-n)x/2 + n is infinite, because the values of n belong to the real set
of numbers.

c) To discover a member that belongs to both
families of linear functions, we'll have to impose the following constraint: the slopes
must be equal and the y intercepts must be also equal.

2 =
(1 - n)/2

4 = 1 - n

n = 1 -
4

n =
-3

Therefore, the member that belongs to both
families of linear functions is y = 2x - 3.