To find the value of [tex]\( n \)[/tex] given that [tex]\( f(x) \)[/tex] is a linear function and the pairs of [tex]\( x \)[/tex] and [tex]\( f(x) \)[/tex] values are provided, follow these steps:
1. Identify the given points: We are given three points with coordinates [tex]\((-4, -25)\)[/tex], [tex]\((-1, -10)\)[/tex], and [tex]\((n, 20)\)[/tex].
2. Formulate the equation of the linear function: The general form of a linear function is [tex]\( f(x) = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
3. Calculate the slope [tex]\( m \)[/tex]: Use the first two points [tex]\((-4, -25)\)[/tex] and [tex]\((-1, -10)\)[/tex] to calculate the slope [tex]\( m \)[/tex].
[tex]\[
m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} = \frac{-10 - (-25)}{-1 - (-4)} = \frac{-10 + 25}{-1 + 4} = \frac{15}{3} = 5
\][/tex]
4. Determine the y-intercept [tex]\( c \)[/tex]: Use the point [tex]\((-4, -25)\)[/tex] and the slope calculated to find [tex]\( c \)[/tex].
[tex]\[
f(x) = mx + c \Rightarrow -25 = 5(-4) + c \Rightarrow -25 = -20 + c \Rightarrow c = -5
\][/tex]
Now, the equation of the linear function is:
[tex]\[
f(x) = 5x - 5
\][/tex]
5. Find the value of [tex]\( n \)[/tex]: Use the point [tex]\( (n, 20) \)[/tex] with the equation [tex]\( f(x) = 5x - 5 \)[/tex] to find [tex]\( n \)[/tex].
[tex]\[
20 = 5n - 5 \Rightarrow 20 + 5 = 5n \Rightarrow 25 = 5n \Rightarrow n = \frac{25}{5} = 5
\][/tex]
Therefore, the value of [tex]\( n \)[/tex] is [tex]\( \boxed{5} \)[/tex].
