Question

The table above gives values of the function $f$ for several values of $t$. If the graph of $f$ is a line, which of the following defines $f\left(t\right)$?

• A. $f\left(t\right)=t-1$
• B. $f\left(t\right)=t+1$
• C. $f\left(t\right)=2t+1$
• D. $f\left(t\right)=2t-1$
• E. $f\left(t\right)=1-2t$

Answer 1

The correct option is D $f\left(t\right)=2t-1$

Graph of the $f$ is line.

Let the equation of the line is

$f\left(t\right)=at+b$ where $a$ and $b$ are some constants.

So when $t=0$, $f=-1$

$\Rightarrow f\left(0\right)=-1$

$\Rightarrow a\times \left(0\right)+b=-1$

$\Rightarrow b=-1$

So when $t=1$, $f=1$

$\Rightarrow f\left(1\right)=1$

$\Rightarrow a\left(1\right)+b=1$

we know that $b=-1$

So, $\Rightarrow a+(-1)=1$

$\Rightarrow a=2$

Checking:

When $t=2$, $f=3$

$\Rightarrow f\left(2\right)=3$

$\Rightarrow 2\times 2-1=3$

$\Rightarrow 3=3$

When $t=3$, $f=5$

$\Rightarrow f\left(3\right)=5$

$\Rightarrow 2\times 3-1=5$

$\Rightarrow 5=5$

The equation $f\left(t\right)=2t-1$ is satisfying the table.

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