Answer :
Given:
A linear function f with f(0)=2 and f(2)=4.
To find:
The linear function.
Solution:
If [tex]f(x)=y[/tex], then function passes through the point (x,y).
A linear function f with f(0)=2 and f(2)=4. It means, the function passes through (0,2) and (2,4). So, the equation of linear function is
[tex]y-y_1=\dfrac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]
[tex]y-2=\dfrac{4-2}{2-0}(x-0)[/tex]
[tex]y-2=\dfrac{2}{2}(x)[/tex]
[tex]y-2=x[/tex]
Adding 2 on both sides, we get
[tex]y=x+2[/tex]
The function form of the equation is
[tex]f(x)=x+2[/tex]
Therefore, the required linear function is [tex]f(x)=x+2[/tex].
The linear function f with f(0)=2 and f(2)=4 is:
f(x) = x + 2
The given points are:
f(0) = 2 and f(2) = 4
That is, x₁ = 0, y₁ = 2, x₂ = 2, y₂ = 4
The slope of the linear function is calculated as shown below
[tex]m = \frac{y_2-y_1}{x_2-x_1} \\\\m = \frac{4-2}{2-0} \\\\m = \frac{2}{2} \\\\m = 1[/tex]
The point-slope form of a linear equation is given as:
[tex]y - y_1 = m(x - x_1)[/tex]
Substitute x₁ = 0, y₁ = 2 and m = 1 into the point-slope equation above
[tex]y - 2 = 1(x - 0)\\\\y-2 = x\\\\y = x + 2[/tex]
The linear function is:
f(x) = x + 2
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