Question

Find the linear function $f$ such that $f\left(2\right)=5$ and $f\left(3\right)=-5$

• A. $f\left(x\right)=2x+1$
• B. $f\left(x\right)=-10x-5$
• C. $f\left(x\right)=3x-1$
• D. $f\left(x\right)=-10x+25$
• E. $f\left(x\right)=-10x-25$

Answer 1

Answer: (D)

$f\left(x\right)=-10x+25$

A linear function $f$ has the form:

$f\left(x\right)=mx+b$

Use $f\left(2\right)=5$ and $f\left(3\right)=-5$ to write

$f\left(2\right)=2m+b=5$ and $f\left(3\right)=3m+b=-5$

Solve the system of equations $2m+b=5$ and $3m+b=-5$.

Subtract both sides to eliminate $b$ as follows:

$\Rightarrow (3m+b)-(2m+b)=-5-5$

$\Rightarrow m=-10$
$\Rightarrow b=5-2m=5-2(-10)$ $=5+20=25$

$\Rightarrow f\left(x\right)=-10x+25$

Hence, option D is correct.