# (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 linear functions with slope 2 and several members of...

(b.) Find an equation for the family of linear functions such that f(2) = 1 and several family members.

(c) Which function belongs to both families?

please need a precise way to solve any sort of these question ??

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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.**