Question

Write a linear function $f$with$f\left(2\right)=-2$ and $f\left(1\right)=1$.

Answer 1

Find linear function.

From $f\left(2\right)=-2$, the point is $(2,-2)$.

From $f\left(1\right)=1$, the point is $(1,1)$.

Recall the formula of two-point form of equation:

$y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$

Substitute the points $(2,-2)$ and $(1,1)$:

$$\begin{gathered} y-1=\frac{-2-1}{2-1}\left(x-1\right) \\ \Rightarrow y-1=\frac{-3}{1}\left(x-1\right) \\ \Rightarrow y-1=-3x+3 \\ \Rightarrow y=-3x+3+1 \\ \Rightarrow y=-3x+4 \end{gathered}$$

Replace $y$ with $f\left(x\right)$:

$f\left(x\right)=-3x+4$

Hence, the linear function is $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{3}\mathit{x}\mathbf{+}\mathbf{4}\mathbf{.}$