Chapter 3.9, Problem 6OE

To determine

To find: The missing entries by value of linear function $f$ .

Answer to Problem 6OE

Therefore, the equation of linear function $f\left(x\right)=3x-1$ and $f\left(0\right)=-4,f\left(-3\right)=5$ .

Explanation of Solution

Given information: The function $f\text{ at }x=-1,x=-2$ ,

  $f\left(-1\right)=-1,f\left(-2\right)=2$

Formula used: For calculation of equation of linear function,

  $\begin{array}{l}f\left(x\right)=mx+b\\ \text{where }m\text{ is slope and b is }y-\text{intercept}\text{.}\end{array}$

Calculation: To calculate the equation of function using function $f\text{ at }x=-1\text{ and }f\text{at }x=-2$ ,

  $\begin{array}{l}\text{Putting value of }f\left(-1\right)\text{in genral form,}\\ f\left(x\right)=mx+b\\ \text{Now, at }x=-1\\ f\left(-1\right)=m\times \left(-1\right)+b\\ -1=-m+b\\ b=-1+m\\ \text{Putting value of }f\left(-2\right)\text{in genral form,}\\ f\left(x\right)=mx+b\\ \text{Now, at }x=-2\\ f\left(-2\right)=m\times \left(-2\right)+b\\ 2=-2m+b\\ \text{Putting value of }b\\ 2=-2m+(-1+m)\\ m=-3\\ \text{Now,}b=-1+m\\ b=-4\\ f\left(x\right)=-3x-4\end{array}$

  $\begin{array}{l}\text{At }x=0,\\ f\left(0\right)=-3\times 0-4\\ f\left(0\right)=-4\\ \text{At }x=-3,\\ f\left(-3\right)=-3\times \left(-3\right)-4\\ f\left(-3\right)=5\end{array}$

Therefore, the equation of linear function $f\left(x\right)=3x-1$ and $f\left(0\right)=-4,f\left(-3\right)=5$ .